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J.N. Newman. Marine Hydrodynamis - 40-th Anniversary Edition

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Marine Hydrodynamics
Marine Hydrodynamics
40th anniversary edition
J. N. Newman
foreword by John Grue
The MIT Press
Cambridge, Massachusetts
London, England
© 2017 Massachusetts Institute of Technology
All rights reserved. No part of this book may be reproduced in any form by any
electronic or mechanical means (including photocopying, recording, or information
storage and retrieval) without permission in writing from the publisher.
This book was set in ITC Stone Serif Std and ITC Stone Sans Std by Toppan Best-set
Premedia Limited. Printed and bound in the United States of America.
Library of Congress Cataloging-in-Publication Data
Names: Newman, J. N. (John Nicholas), 1935– author.
Title: Marine hydrodynamics / J. N. Newman ; foreword by John Grue.
Description: 40th anniversary edition. | Cambridge, MA : The MIT Press,
[2017] | Includes bibliographical references and index.
Identifiers: LCCN 2017023272 | ISBN 9780262534826 (pbk. : alk. paper)
Subjects: LCSH: Ships--Hydrodynamics. | Hydrodynamics.
Classification: LCC VM156 .N48 2017 | DDC 623.8/12--dc23 LC record available at
https://lccn.loc.gov/2017023272
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To my family and friends
Contents
Foreword
xi
Preface to the 40th Anniversary Edition
Preface to the First Edition
1 Introduction
xvii
xix
1
2 Model Testing
9
2.1 Falling Body in a Vacuum
2.2 Pendulum
10
11
2.3 Water Waves
12
2.4 Drag Force on a Sphere
14
2.5 Viscous Drag on a Flat Plate
17
2.6 Viscous Drag on General Bodies
2.7 Hydrofoil Lift and Drag
2.8 Screw Propeller
18
22
25
2.9 Drag on a Ship Hull
29
2.10 Propeller-Hull Interactions
34
2.11 Unsteady Force on an Accelerating Body
2.12 Vortex Shedding
40
2.13 Wave Force on a Stationary Body
2.14 Body Motions in Waves
45
2.15 Ship Motions in Waves
48
Problems
References
41
50
53
3 The Motion of a Viscous Fluid
3.1 Description of the Flow
55
56
3.2 Conservation of Mass and Momentum
58
37
viii
Contents
3.3 The Transport Theorem
59
3.4 The Continuity Equation
3.5 Euler's Equations
61
62
3.6 Stress Relations in a Newtonian Fluid
3.7 The Navier-Stokes Equations
3.8 Boundary Conditions
62
65
66
3.9 Body Forces and Gravity
66
3.10 The Flow between Two Parallel Walls (Plane Couette Flow)
3.11 The Flow through a Pipe (Poiseuille Flow)
3.12 External Flow Past One Flat Plate
3.13 Unsteady Motion of a Flat Plate
68
70
72
3.14 Laminar Boundary Layers: Steady Flow Past a Flat Plate
3.15 Laminar Boundary Layers: Steady Two-Dimensional Flow
3.16 Laminar Boundary Layers: Closing Remarks
3.17 Turbulent Flow: General Aspects
88
88
3.18 Turbulent Boundary Layer on a Flat Plate
3.19 The 1/7-Power Approximation
91
99
3.20 Roughness Effects on Turbulent Boundary Layers
3.21 Turbulent Boundary Layers: Closing Remarks
Problems
References
100
102
102
104
4 The Motion of an Ideal Fluid
4.1 Irrotational Flows
107
108
4.2 The Velocity Potential
110
4.3 Bernoulli's Equations
112
4.4 Boundary Conditions
114
4.5 Simple Potential Flows
4.6 The Stream Function
4.7 The Complex Potential
4.8 Conformal Mapping
4.9 Separation of Variables
116
121
123
125
129
4.10 Fixed Bodies and Moving Bodies
132
4.11 Green's Theorem and Distributions of Singularities
4.12 Hydrodynamic Pressure Forces
67
133
138
4.13 Force on a Moving Body in an Unbounded Fluid
4.14 General Properties of the Added-Mass Coefficients
141
147
75
81
Contents
ix
4.15 The Added Mass of Simple Forms
4.16 The Body-Mass Force
151
155
4.17 Force on a Body in a Nonuniform Stream
4.18 The Method of Images
Problems
References
156
160
161
164
5 Lifting Surfaces
167
5.1 Two-Dimensional Hydrofoil Theory
169
5.2 Linearized Two-Dimensional Theory
5.3 The Lifting Problem
172
176
5.4 Simple Foil Shapes
180
5.5 Drag Force on a Two-Dimensional Foil
184
5.6 Two-Dimensional Source and Vortex Distributions
5.7 Singular Integral Equations
189
5.8 Three-Dimensional Vortices
197
5.9 Three-Dimensional Planar Lifting Surfaces
5.10 Induced Drag
206
5.11 Lifting-Line Theory
5.12 Cavity Flows
210
216
5.13 Symmetric Cavity Flows
218
5.14 Supercavitating Lifting Foils
5.15 Unsteady Hydrofoil Theory
223
229
5.16 Oscillatory Time Dependence
5.17 The Sinusoidal Gust Problem
5.18 Transient Problems
Problems
References
236
239
241
242
244
6 Waves and Wave Effects
247
6.1 Linearized Free-Surface Condition
6.2 Plane Progressive Waves
6.3 Finite-Depth Effects
6.4 Nonlinear Effects
6.5 Mass Transport
250
253
256
261
6.6 Superposition of Plane Waves
6.7 Group Velocity
6.8 Wave Energy
268
271
263
248
200
186
x
Contents
6.9 Two-Dimensional Ship Waves
277
6.10 Three-Dimensional Ship Waves
282
6.11 The Method of Stationary Phase
286
6.12 Energy Radiation and Wave Resistance
290
6.13 Thin-Ship Theory of Wave Resistance
6.14 Wave Pattern Analysis
6.15 Body Response in Regular Waves
6.16 Hydrostatics
292
294
297
302
6.17 Damping and Added Mass
306
6.18 Wave-Exciting Force and Moment
313
6.19 Motion of Floating Bodies in Regular Waves
6.20 Ocean Waves
6.21 Motions of Bodies in Irregular Waves
Problems
References
333
334
338
7 Hydrodynamics of Slender Bodies
341
7.1 Slender Body in an Unbounded Fluid
7.2 Longitudinal Motion
7.3 The Lateral Force
342
348
351
7.4 Ship Maneuvering: The Hydrodynamic Forces
7.5 Ship Maneuvering: The Equations of Motion
7.6 Slender Bodies in Waves
7.8 Slender Bodies in Shallow Water
References
357
363
369
7.7 Strip Theory for Ship Motions
Problems
319
323
374
388
397
399
Appendix: Units of Measurement and Physical Constants
Notes
405
Index
409
403
Foreword
F
John
o
r
e
w
o
r
d
Grue
© Massachusetts Institute of TechnologyAll Rights Reserved
I became acquainted with this book in 1980 while completing my master’s
degree in mechanics at the University of Oslo. After the North Sea oil boom
of the 1970s, it was apparent that there was a need for improved higher
education and research related to the oil industry. The University of Oslo
developed a curriculum in marine hydrodynamics, and interest in the field
grew rapidly as students signed up for the program. I was one of them.
The course in marine hydrodynamics was offered for the first time during the 1980–1981 academic year, and has been lectured approximately
once a year since its inception. J. N. Newman’s textbook felt like a gift to
our cohort and to our professor, Enok Palm. The entire book comprised a
new curriculum with the exception of chapter 3, as viscous flow was covered in other courses. Like us, Palm was new to the subject, so the course
was taught as a seminar. I was assigned the duty of giving lectures based
on the book because our professor believed that having the youngest (and
presumably the least experienced) student give the lectures would slow the
pace of the course. I presented two hours of lectures each week for the entire
academic year. During the lectures, Palm conducted a detailed examination
of all statements and deductions I wrote on the blackboard, and I fielded
questions from my classmates as well. By the end of the spring term, I knew
the book by heart. The following year, the entire research division of Det
Norske Veritas (now DNV GL) attended the course. This time, Enok Palm
lectured the course himself. “I just can’t ask a master’s student to teach such
an important course to the scientific leaders of Veritas,” he told me. I was
given the duty of leading the exercises instead.
Reading this book—a real classic—has been the most defining experience of my scientific career. I use it frequently in my teaching and scientific
xii
Foreword
work. This text is a continuous font of inspiration for a master’s or PhD
project.
How does one define marine hydrodynamics? First, it’s important to
understand the meaning of offshore engineering. Put simply, offshore
engineering deals with the design, construction, and safety considerations,
including insurance, of both stationary and moving structures at sea.
Regarding the first question, marine hydrodynamics defines the theoretical
framework of offshore engineering. Marine hydrodynamics is an extensive
subject as it treats the hydrodynamics in relation to all conceivable geometries exposed to the forces in the ocean environment. This includes the
forces from the waves as well as those acting on lifting surfaces, either in
water or in air. Sailors, rowers, canoeists, and kayakers are all faced with
the same challenges: how does one reduce wave or frictional resistance,
and optimize sail, rudder shape, or paddle motion? These are all examples
of marine hydrodynamics. Calculating extreme loads and thereby ensuring the survival of structures exposed to extreme conditions—such as the
record high Draupner wave, detected in the North Sea on New Year’s Day
1995 with a crest height of more than 18 meters, a wave height of more
than 25 meters, and a wavelength slightly exceeding 200 meters—is the
ultimate goal. (Note that the design waves of the platforms in the North Sea
are stronger than the Draupner wave.)
Stationary bodies are either fixed to the seafloor or floating on the surface with a stiff or slack mooring maintaining their position. There is a
variety of such bodies, including all kinds of ships, barges, offshore platforms in transit from one position to another, and other objects towed
by ships. Offshore wind turbines are more recently studied geometries. In
most cases these are organized in large wind farms that are mounted on
the seafloor as at Horns Rev 1 and 2 in Danish waters, and at Dudgeon on
the coast of the United Kingdom. Hywind, developed by the Norwegian
oil and gas company Statoil, is an offshore wind turbine prototype that
floats on a spar buoy. The current target wing span of these geometries
is twice that of an Airbus A380! Other geometries in offshore engineering include cages for aquaculture, which may be either closed or open.
Making them larger and sufficiently strong enough to be placed in harsh
ocean environments is a new development in the fish farming industry.
New development of coastal infrastructure would require road crossings of
fjords. Proposed bridges would be supported by floating, moored pontoons;
John Grue
xiii
submerged floating tunnels are another alternative. Wave power devices
are another example of structures where the subject of marine hydrodynamics is relevant, and where the design of successful structures must be
based on this subject.
This textbook is ideal for a masters- or PhD-level course. Students using
this book should also have completed courses in introductory mathematics,
classical physics, and field theory. Additionally, a course in fluid mechanics
provides a foundation to build upon with the material contained herein.
Most introductory courses in mathematics have some components of complex theory and provide a sufficient background for the more advanced
exercises in the book; a full course in complex analysis may help the reader
but is not required for a good learning outcome.
The highlight of the book is the second half of chapter 6, which introduces the subject of wave effects. The primary goal is to calculate the
responses of a floating body exposed to incoming waves in the six degrees
of freedom: heave, sway, surge, roll, pitch, and yaw. Chapter 6, much like
the rest of the book, is self-consistent, with the required background of
wave theory—including concepts like superposition, group velocity, mass
and energy flux, and third-order Stokes wave theory—given in the first half
of the chapter. On the basis of potential theory, the matrix equation of the
body responses includes the forces and moments due to added mass, wave
damping, hydrostatics, and wave excitation. Perturbation-wise, the wavebody interaction problem is linearized and the forces and motions are modeled at each spectral frequency, a common assumption and decomposition
that is used in marine hydrodynamics. The method provides a platform for
a rich mathematical analysis in which the wave effects and responses are
analyzed at the fundamental frequency. The resonance frequencies in the
vertical modes of motion where the hydrostatic forces balance the inertia
forces are defined. The linear analysis corresponds to the basic calculation
method used in offshore engineering. Quadratic and higher-order wave
effects may be studied using other texts.
The introduction of Morison’s equation—a coefficient-based force formulation, particularly suitable for slender geometries—is motivated by the
dimensional analysis in chapter 2. It is directly related to the long-wave
analysis of the wave effects part of chapter 6. The viscous drag term included
in Morison’s equation is not modeled by the potential theory analysis of
chapter 6, however. Chapter 2 introduces the concept of added mass—an
xiv
Foreword
inertia effect due to the accelerated mass of the surrounding fluid. It contributes in a similar way to the mass of a body in classical mechanics. In
chapter 4—also directly related to the wave effects part of chapter 6—the
forces and moments on a geometry moving in the six degrees of freedom,
in unbounded fluid, are derived on exact form, expressed in terms of the
six-by-six added mass matrix. This elegant, exact analysis is directly suitable as a model of the motion of a remotely operated underwater vehicle
(ROV).
The ways in which animals swim and fly, airplanes and ski jumpers control their flight, and a yacht sails across the water are the result of flow at
thin streamlined bodies. Chapter 5 introduces the fundamentals of the flow
and force on a lifting surface. Moreover, it describes the propulsive force
created by a wing’s flapping motion. The lift problem of a two-dimensional
foil section is analyzed in the beginning of the chapter—a starting point
in the training of candidates for the growing wind power industry. The
sharp trailing edge of the wing enforces the flow velocity to be finite,
imposed mathematically by the Kutta condition, controlling the initial
flow separation.
The mathematical modeling in this book is represented by partial differential equations (PDEs). The solution methods of field equations in marine
hydrodynamics—commonly the Laplace equation—differ fundamentally
from the common solution methods of PDEs. The main reason is this: The
cause of the motion (the geometry) is local whereas the motion is global
(the infinite extension of the wave field). Popular methods for solving
partial differential equations are often based on volume methods, either
difference methods or variants of the finite element method. The special
method of separation of variables employs sets of orthogonal functions
particularly fitted to geometries of special shape. However, for floating or
submerged large-volume geometries of arbitrary shape, use of integral equations is unsurpassed and is a standard method in marine hydrodynamics.
This method is extensively used in the offshore industry. The formulation,
in terms of integral equations, is directly related to the scattering and radiation problems in theoretical physics, although the wave Green functions
differ. The application of multipole expansions including the surface wave
effects is an efficient, accurate, and convergent method that has found a
lesser application in the industry. Thus, chapter 4 involves solving PDEs
for the flow in an unbounded fluid by use of integral equations expressed
John Grue
xv
in terms of source and dipole distributions. In chapter 5, the lifting and
flapping problems are modeled by vortex and source distributions. The
resulting integral equation becomes a singular Fredholm integral equation
of the first kind. The equation is directly invertable—a superior, analytical,
and generic method directly transferable to other related formulations and
problems. The nonsingular integral equation formulated in section 11 of
chapter 4 is a Fredholm equation of the second kind. This formulation is
robust and easy to invert. Variants of this equation are widely used in the
offshore industry.
How do I teach this book? I start with the integral representation of the
potential flow at a geometry in unbounded fluid found in section 4.11, and
subsequently introduce the forces and moments in an unbounded fluid
expressed in terms of the added mass matrix. The first assignment is the
calculation of the two-dimensional added mass of a circle, an ellipse, and
a square, obtaining the various potentials through integral equations and
Python or MATLAB scripting. Problems 4, 6, 13, and 15 are mandatory.
I continue with chapter 6. I usually give one lecture on linear wave theory and a second lecture on energy flux and group velocity. The wave effects
part of the chapter comes next, introducing the decomposition of the diffraction and radiation potentials, the latter according to the six modes of
motion, the added mass and damping, the exciting forces, and the restoring
forces. Sometimes, depending on the student group, I will assign section
6.16 (hydrostatics) separately. The students then derive the mathematical
formulas in the chapter for subsequent presentation in class. The theories
are reworked by completing problems 10–17 on wave effects. Problems 1–8
cover the fundamentals of wave theory.
In chapter 5, my lecture covers the introduction and sections 5.1–5.6, as
well as problems 8 and 9. The results are put in context of the dimensional
analysis of the foil in section 7 of chapter 2.
One of my final lectures covers the dimensional analysis of chapter
2, with highlights including the low-level introduction of the concept of
added mass and the coefficient-based force formulation included in Morison’s equation. Problem 14 illustrates how the contributions from the
inertia term and the drag term depend on the cylinder diameter. Froude’s
hypothesis for the drag on a ship hull is one of my favorites; I frequently use
the ITTC-line in figure 2.12 to estimate the frictional resistance on ships.
I used this curve as well as equation (23) of chapter 2 when advising the
xvi
Foreword
kayaker Eirik Verås Larsen as he prepared for the 2012 London Olympics.
His question was how to reduce frictional and wave resistance. I suggested
that the only variable to play with was the wet area of the kayak—the portion touching the water, which should be minimal. Larsen and his team
eventually discovered that he could reduce his weight—which he eventually reduced by 10 percent—and won the gold medal on the K1 1000m
event.
John Grue
University of Oslo
March 16, 2017
Preface to the 40th Anniversary Edition
Preface
Preface
to
to
the
the
40th
40th
Anniversary
Anniversary
Edition
Edition
© Massachusetts Institute of TechnologyAll Rights Reserved
The field of marine hydrodynamics has broadened greatly over the past
40 years, with applications to a wide variety of vessels and structures.
These include systems for converting energy from the wind, waves, and
currents; yachts, high-speed vessels, aquaculture facilities, and various
types of submerged vessels; and traditional applications to ships and offshore platforms. The support for education and research has grown accordingly; it is gratifying that the term “marine hydrodynamics” has become
ubiquitous for university departments, research laboratories, conferences,
and publications.
The basic topics of the field are unchanged, corresponding broadly to
the chapters of this text. Numerical methods that extend the applications
of the theory have been developed. Practicing engineers and naval architects are now making routine use of well-established programs to optimize
their designs and predict performance. These programs include Navier–
Stokes solvers, which analyze viscous effects including turbulence, and
“panel” programs based on boundary-integral equations to solve potentialflow problems including lifting and wave effects. This evolution has been
accelerated by the universal access to computers of increasing capacity
and convenience. Nevertheless, it is essential to understand the underlying
principles covered in this text, and to compare the results of computations
with simpler approximations to be confident of their validity.
I am grateful to the MIT Press for suggesting this special edition and
for making it available economically, both as a paperback and as an open
access e-book. I am especially grateful to Professor John Grue for the foreword, which reflects his long experience using this text.
Preface to the First Edition
P
P
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e
f
f
a
a
c
c
e
e
© Massachusetts Institute of TechnologyAll Rights Reserved
The applications of hydrodynamics to naval architecture and ocean
engineering have expanded dramatically in recent years. Ship design has
been related increasingly to the results of scientific research, and a new
field of ocean engineering has emerged from the utilization of offshore
resources. The number of technical symposia and journals has increased
in proportion to this expansion, but the publication of textbooks has not
kept pace.
This volume has been prepared to satisfy the need for a textbook on
the applications of hydrodynamics to marine problems. These pages have
evolved from lecture notes prepared for a first-year graduate subject in the
Department of Ocean Engineering at MIT, and used subsequently for undergraduate and graduate courses at several other universities. Most of the students involved have taken an introductory course in fluid mechanics, but
the necessary fundamentals are presented in a self-contained manner. A
knowledge of advanced calculus is assumed, including vector analysis and
complex-variable theory.
The subject matter has been chosen primarily for its practical importance, tempered by the limitations of space and complexity that can be tolerated in a textbook. Notably absent are topics from the field of numerical
hydrodynamics such as three-dimensional boundary-layer computations,
lifting-surface techniques including propeller theory, and various numerical solutions of wave-body problems. A textbook on these subjects would
be a valuable companion to this volume.
Since most countries of the world have adopted the rationalized metric
Système International d'Unités (SI), this is used here except for occasional
references to the “knot” as a unit of speed. Conversion factors for English
xx
Preface
units of measure are given in the appendix, together with short tables of
the relevant physical properties for water and air. A unified notation has
been adopted, despite the specialized conventions of some fields. Cartesian
coordinates are chosen with the y-axis directed upward. Forces, moments,
and body velocities are defined by an indicial notation that differs from
the standard convention of ship maneuvering. The symbol L is reserved for
the lift force, and D for drag. Thus length is denoted by l and diameter by
d. Vessels with a preferred direction of forward motion are oriented toward
the positive x-axis, following the practice of naval architecture but contrary
to the usual convention of aerodynamics; a fortunate consequence is that a
hydrofoil with upward lift force will possess a positive circulation as defined
in the counterclockwise sense.
This text was initiated with the enthusiastic encouragement of Alfred H.
Keil, Dean of Engineering at MIT, and Ira Dyer, Head of the Department of
Ocean Engineering. Financial support has been provided by the Office of
Naval Research Fluid Mechanics Program, which for the past thirty years
has fulfilled an invaluable role in the development of this field. Additional
thanks are due to the National Science Foundation and the David Taylor
Naval Ship Research and Development Center for their support of the
research activities that have filtered down into these pages.
Many colleagues and former students have helped significantly with
encouragement, advice, and assistance. John V. Wehausen of the University of California, Berkeley, pioneered in applying the discipline of
contemporary fluid mechanics on a broad front to the teaching of naval
architecture; he has been generous with his advice as well as his own
extensive lecture notes. Justin E. Kerwin of MIT shared in developing the
course from which this text has evolved, and he has been particularly
helpful in discussing a broad range of topics. Other colleagues to whom
I am indebted include Chryssostomos Chryssostomidis, Edward C. Kern,
Patrick Leehey, Chiang C. Mei, Jerome H. Milgram, Owen H. Oakley, Jr.,
and Ronald W Yeung of MIT; Keith P. Kerney, Choung M. Lee, and Nils
Salvesen of the David Taylor Naval Ship Research and Development Center; Robert F. Beck and T. Francis Ogilvie of the University of Michigan; T.
Yao-tsu Wu of the California Institute of Technology; Odd Faltinsen of the
Norwegian Technical University; P. Thomas Fink of the University of New
South Wales; and Ernest O. Tuck of the University of Adelaide. Former
Preface
xxi
students who have been particularly helpful in a variety of ways include
Elwyn S. Baker, Charles N. Flagg, George S. Hazen, Ki-Han Kim, James H.
Mays, and Paul J. Shapiro.
The original illustrations are from the talented pen of Lessel Mansour.
The manuscript was typed with proficiency by Jan M. Klimmek, Jacqueline
A. Sciacca, and Kathy C. Barrington. My wife Kathleen helped with many
editorial tasks and has patiently endured the diversion of my time.
1
Introduction
Chapter
I
n
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© Massachusetts Institute of TechnologyAll Rights Reserved
The applications of hydrodynamics to naval architecture and ocean engineering cover many separate topics and range over a broad level of sophistication. The topics are as diverse as the propulsion and steering of ships
and the behavior in waves of a moored buoy or oil-drilling platform.
The former are classical problems of naval architecture, predated only
by Archimedean hydrostatics. The buoy and platform problems are more
recent from the standpoint of scientific and engineering analyses. The
degree of sophistication varies from empirical design methodology to theoretical research activities whose justification is based on long-range hopes
of application.
The fields of technology are also diverse, and to solve these problems
requires a knowledge not only of fluid mechanics but also of solid mechanics (to describe the mooring system especially), control theory (to represent
the mechanical and human systems involved), as well as statistics and random processes (to deal with the highly irregular environment of the ocean).
Here we shall focus our attention specifically on the hydrodynamic aspects,
emphasizing those unique to this field as opposed to the other engineering
disciplines where fluid mechanics is applicable.
Faced with the choice between empirical design information and esoteric
theory, we will follow a middle course to provide the necessary background
for an intelligent evaluation and application of empirical procedures and
also serve as an introduction to more specialized study on the research end
of the spectrum. This approach has the advantage of being a compromise
between two viewpoints, which sometimes appear to conflict; it also unifies the seemingly diverse problems of marine hydrodynamics by examining them not as separate problems but instead as related applications
of the general field of hydrodynamics. For example, propellers, rudders,
2
Chapter 1
antirolling fins, yacht keels, and sails are all fundamentally related to
wings and hydrofoils, or lifting surfaces, and can be treated and understood
together. Similarly, the unsteady ship, buoy, or platform motions in waves
and the maneuvering of ships or submarines in nonstraight paths can be
analyzed, to some extent, from the same basic equations of motion. In fact,
however, the maneuvering problem generally involves separation and lifting effects, whereas the motions of bodies in waves are not as significantly
affected by viscosity or vorticity.
The dynamics of fluid motions, like the dynamics of rigid bodies, are
governed by the opposing actions of different forces, and moments, which
are implied when not explicitly included. In fluid dynamics, these forces
can no longer be considered as acting at a single point or discrete points
of the system; instead they must be distributed in a relatively smooth or
continuous manner throughout the mass of fluid particles. The force distribution and the kinematic description of the fluid motion are in fact continuous if and only if we assume that the discrete molecules of fluid can be
analyzed as a continuum.
Typically, we can anticipate force mechanisms associated with the fluid
inertia, its weight, viscous stresses, and secondary effects such as surface
tension. In general, the three principal force mechanisms—inertial, gravitational, and viscous—are of comparable importance. With very few exceptions, it is not possible to analyze such a complicated situation, either
theoretically or experimentally, and we can either give up or try to assess
intelligently the role of each force mechanism, in the hope of subsequently
treating them in pairs. As we shall see, this relatively simple state of affairs
is still fraught with difficulties.
It is useful first to estimate the orders of magnitude of the inertial, gravitational, and viscous forces. We shall suppose that the problem at hand can
be characterized by a physical length l, velocity U, fluid density ρ, gravitational acceleration g, and a coefficient of fluid viscosity μ. We then can
estimate the three forces:
Type of Force
Order of Magnitude
Inertial
ρU2l2
Gravitational
ρgl3
Viscous
μUl
Introduction
3
These estimates should not be interpreted too strictly. For example, it
could be argued, from Bernoulli’s equation, that a factor of
1
2
should be
associated with the magnitude of the inertial force. Nevertheless, the estimates are valid in the sense that changes in the magnitudes of any of the
physical parameters l, U, ρ, g, or μ, will affect the forces as indicated. Thus,
suppose the length scale is doubled, as might follow from attempts to compare directly the forces acting on a 100 m ship and on a 200 m ship, moving with the same speed; then the corresponding changes in the inertial,
gravitational, and viscous forces will be multiplicative factors of 22, 23, and
2, respectively. Therefore, the fundamental balance among the three types
of force will change as the length scale changes, and the effects will be
more pronounced if we anticipate the relatively large changes of length scale
(on the order of ten or one hundred) associated with a comparison of
small-scale models and full-scale vessels.
To predict full-scale phenomena from tests with a scale-model, the
absolute magnitude of any one force acting alone could be corrected by a
suitable multiplicative factor. The principal concern is that all three forces
act simultaneously and that their relative magnitudes be preserved so that
the resulting flow is dynamically similar. For this reason, it is illuminating
to form the ratios of the three forces, which yield a set of three nondimensional parameters to describe the fluid flow:
Inertial Force
ρU 2l 2
=
= U 2 /gl ,
Gravitational Force
ρ gl 3
Inertial Force ρU 2l 2
=
= ρUl /µ ,
Viscous Force
µUl
Gravitational Force ρ gl 3
=
= ρ gl 2 /µU .
Viscous Force
µUl
Any two of these three ratios are sufficient to define the third and hence to
determine the balance of forces in the fluid motion. Customarily, the first
two are employed in the forms
F = Froude Number = U /(gl )1/2 ,
R = Reynolds Number = ρUl /µ = Ul /ν ,
where ν = μ/ρ is the kinematic viscosity coefficient of the fluid. A short
table of the density and viscosity coefficients for water and air is given in
the appendix. Typical values for the kinematic viscosity ν are 10−6 m2/s (10−5
4
Chapter 1
ft2/s) for water and 1.5 × 10−5 m2/s (1.5 × 10−4 ft2/s) for air. That this coefficient is small when expressed in terms of conventional units implies that
the Reynolds number R will be large; hence viscous forces will be negligible
relative to inertial forces. This is generally true, but one must be more cautious before concluding that viscosity can be completely ignored. In fact,
viscosity can be neglected for the bulk of the fluid but must be included in
singular regions such as the boundary layer very close to a body.
In this analysis we have tacitly assumed steady motion and hence constant characteristic velocity U. If instead the motion is oscillatory in time, as
in the case of a buoy oscillating in a seaway, then the characteristic velocity
U should be replaced by the combination ωl, where ω is the frequency of
oscillations in radians per unit time. The counterpart of the Froude number for such motions is the nondimensional frequency parameter ω(l/g)1/2.
Alternatively, we can use the ratio λ/l, where λ is the wavelength, or distance between successive wave crests, since λ can be explicitly related to ω
(λ = 2πg/ω2 for waves in deep water).
Other important parameters arise when we examine the kinematic and
thermodynamic aspects of the fluid. The most familiar of these is the Mach
number of aerodynamics, which is the ratio between the velocity U and
the speed of sound in the fluid medium; it arises from considerations of
the elasticity or compressibility of the fluid. These effects are not important in the motions and behavior of ocean vehicles because water is not
significantly compressible at the speeds of interest; in terms of the Mach
number we note that the speed of sound in water is on the order of 1,500
m/s or 3,000 knots, which implies a negligibly small Mach number for
water—based craft and thus insignificant compressibility effects. On the
other hand we must consider in certain circumstances the possibility of
cavitation, for when the pressure in the fluid is reduced below the vapor
pressure pν, the physical state of the fluid abruptly changes to that of a gas.
Thus, while the fluid is extremely inelastic in compression, it cannot normally sustain significant tension. Dimensional considerations suggest the
cavitation number
σ=
p0 − pν
,
2
1
2 ρU
which is a measure of the likelihood of cavitation and parametrically
describes the subsequent details. Here p0 is a characteristic pressure level
Introduction
5
in the fluid, such as the hydrostatic pressure at the depth in question, and
the vapor pressure pν depends on the properties of the fluid and its temperature. If the cavitation number is large, cavitation will not occur, and
the precise value of σ is immaterial to the description and analysis of the
flow. If the cavitation number is sufficiently small so that cavitation occurs
in the flow field, dynamic similarity between two flows will exist only if
the corresponding cavitation numbers are equal. At normal temperatures
pν is substantially less than the atmospheric pressure, and thus cavitation is
significant only at very high speeds.
Fluid motions that have similar geometries but different values of the relevant physical parameters are said to be dynamically similar if the relevant
nondimensional parameters such as the Froude and Reynolds numbers are
equal. It follows that the relative balance between the inertial, viscous, and
gravitational forces is identical, and the resulting hydrodynamic effects,
including fluid velocity, pressure, and forces acting on the boundaries of
the fluid can all be analyzed by means of relationships between the two
different flows. Dynamic similitude is clearly desirable if one is conducting
small-scale model tests that can be used to design large-scale vessels. However, the simultaneous scaling of both Froude and Reynolds numbers is not
possible, at least for reasonable changes of length scale with water as the
full-scale fluid. To see this simply note that the ratio of the Reynolds and
Froude numbers, (g1/2l3/2/ν), must stay constant. For a model length substantially smaller than the full-scale vessel, either the gravitational acceleration must be increased or the viscosity coefficient decreased, by an order of
magnitude. The former suggests a centrifuge and the latter a superfluid, but
neither is amenable to exploitation in this context.
This discussion of dynamic similitude demonstrates that assumptions
or simplifications are often necessary to apply experimental methods with
models to the hydrodynamics of ocean vessels. This is equally true, if not
more so, if one wishes to pursue a strictly analytical prediction based on
rational mechanics. Ultimately, therefore, we must expect both experiments and theory to be used and to be supplemented by full-scale observations to verify the original predictions.
In the theoretical approach the motion of the fluid is defined at each
point in time and space by a kinematic description, usually of the vectorial
velocity of the fluid particles or alternatively of the three scalar components of the velocity. This unknown velocity function is related by means
6
Chapter 1
of Newton’s equations to the forces that act upon the fluid; this yields a
system of partial differential equations. For a fluid with conventional stress
relations, the resulting system of governing equations is the Navier-Stokes
equations, supplemented by the continuity equation expressing conservation of fluid mass. In principle, one can solve these equations, subject to
boundary conditions on the boundary surfaces of the fluid. If this procedure could actually be carried out, it would be possible to calculate desired
answers for arbitrary values of the Reynolds number and Froude number,
and the scaling dilemma of model testing would be circumvented. In practice, however, it has not been possible to solve the Navier-Stokes equations
exactly, except for a few cases involving very simple geometries that at first
glance have no relation to the shape of marine vessels.
Considerably more analytical progress can be made if the viscous forces
are ignored in the Navier-Stokes equations and the fluid is assumed to be
inviscid or ideal. It is then feasible to construct mathematical solutions for
the flow past bodies of realistic form and even to include the effects of wave
motions on the free surface, albeit only after further idealizations. Clearly,
however, neglecting viscosity will lead to results of only academic interest,
unless the justification is more relevant than the mathematical desire to
simplify a system of partial differential equations. For predictions of ship
resistance, this justification was initially provided by Froude’s hypothesis
that the resistance could be composed of two separate components, frictional and residual. The frictional component is related to a much simpler geometry, a deeply submerged flat plate, and thus depends only on
the Reynolds number. The residual component is assumed to depend only
on the Froude number. This hypothesis was essentially an empirical one;
its principal justification was that it led to a workable procedure for making model tests and obtaining full-scale predictions not totally at variance
with the actual full-scale results. The neglect of viscosity in treating certain
aspects of the flow, and Froude’s assumption that the frictional resistance
of a ship hull could be related to that of a flat plate of the same length
and area, found a more rational justification after Prandtl developed the
boundary-layer theory. Thus it became evident that at the relatively large
Reynolds numbers of interest to naval architects and aerodynamicists, viscous stresses are significant only within the very thin layer of the fluid
adjoining the rigid surfaces such as the ship’s hull or the airplane. Outside this layer the fluid is essentially inviscid, not because its viscosity has
Introduction
7
suddenly changed but because viscous stresses are a consequence of large
gradients in the fluid velocity and these large gradients are restricted to the
immediate vicinity of the boundary. Moreover, the flow within the boundary layer is relatively insensitive to the form of the boundary, provided the
body’s radii of curvature are large compared to the thickness of the boundary layer. The consequences of Prandtl’s boundary-layer theory would
appear to be fundamental to Froude’s hypothesis, and it is remarkable that
Froude predated Prandtl by thirty years.
In the chapters to follow, model testing will be discussed first, with
emphasis on the use of dimensional analysis to preserve dynamic similarity
between the model and full-scale flows. This glimpse of the “real-world”
will serve to introduce the theories that follow in subsequent chapters. The
theoretical approach will commence with a study of viscous flows so that
the importance of viscosity can be set in proper perspective.
Each chapter begins with a general treatment of the essentials and progresses to more specialized and advanced material. The order in which this
is studied can be varied to suit one’s interests and background. Readers anxious to proceed to their favorite subject, but lacking the prerequisites to do
so directly, should find most of their needs met in sections 3.1 to 3.9 and
4.1 to 4.5. In the references listed at the end of each chapter, preference has
been given to recent surveys and papers with comprehensive bibliographies
where additional information on each topic may be sought.
2 Model Testing
Chapter
Model
2
Testing
© Massachusetts Institute of TechnologyAll Rights Reserved
For most of us, toy boats and models are a useful childhood introduction
to the subject of hydrodynamics. Given a block of wood, a sharp knife, and
the plans of a ship or small boat, few would resist the impulse to produce
a model. Our choice of scale ratio would be determined by the dimensions
of the block of wood. Perhaps future engineering interests are indicated if
a ruler is used to ensure a “true scale model.” If the model is subsequently
operated in its natural medium, the builder must determine the proper
weight and, in the most refined examples, an appropriate power system
to drive the model at the “true” speed of the real ship. The weight may be
determined by an application of Archimedes’ principle in the bathtub or
sink, but the questions pertaining to speed and power are more complex
and, strictly speaking, impossible to answer. Superficial bathtub observations might reveal that the flow of water around the model is reasonably
like that of a large ship, meaning that the observed waves look realistic, if
the model speed is on the order of half a meter per second, or one knot.
On the other hand, if we could observe the viscous flow very close to the
surface of the hull and in the wake immediately behind the stern, gross differences would be apparent between the ship and its scale model.1
This situation can be better understood by a dimensional analysis.
Denote the physical quantity of interest by Q (for example, the resistance
of a body moving through the water, the thickness of the boundary layer,
or the acceleration of a buoy in waves). First it is necessary to list, from a
qualitative knowledge of the underlying physical mechanisms involved,
the significant parameters that affect the value of Q (length, velocity, density, viscosity, and so forth). In general, some or all of these parameters can
be used to make Q nondimensional, and any not needed for this purpose
can themselves be rendered nondimensional. We then assert that when
10
Chapter 2
expressed in nondimensional form, the quantity of interest must depend
only on the remaining nondimensional parameters. This assertion is equivalent to the statement that the event in question is not affected by the
choice of units of measurement.
We are concerned exclusively with problems where the three fundamental units are mass (M), length (L), and time (T); hence the unknown Q and
the significant parameters upon which it depends can be expressed in terms
of these units. If Q depends on N − 1 significant parameters, there will be a
total of N interrelated dimensional quantities, including Q. The number of
interrelated nondimensional quantities is always smaller, hence the resulting simplification of the problem. Since there are three fundamental units
(M, L, T), the number of independent nondimensional parameters will be
reduced by the same number. Hence a total of N − 3 nondimensional quantities must be interrelated.
This statement is essentially an intuitive one. A careful mathematical
proof is possible; the end result, known as the Pi theorem, is derived by
Birkhoff (1955) and Sedov (1959). Our approach here will be more pragmatic; it will illustrate the procedure of dimensional analysis and the conclusions that follow from the Pi theorem by considering a sequence of
progressively more complicated physical problems.
2.1 Falling Body in a Vacuum
Let us consider first the problem of a body falling freely in a vacuum. The
unknown quantity Q will be taken as the vertical position y, with dimension L. The unknown y might depend on time t, body mass m, and the
gravitational acceleration g which has units L/T2. Since the body is falling
in a vacuum, its size and shape cannot affect the descent. The vertical position y can be nondimensionalized by dividing by the quantity gt2 to form
the unknown y/gt2. However, there is no way of forming the significant
parameters t, m, g into nondimensional parameters, since neither g nor t
contains the units of mass. Thus we conclude that
y / gt 2 = C ,
(1)
where C is a pure constant, a nondimensional function of nonexistent,
nondimensional parameters involved in the problem. The constant C is
Model Testing
11
known from Newtonian mechanics to be equal to 1/2; but if we did not
know this, we could determine C from a single experimental observation,
and dimensional analysis would relate the results of this one experiment to
all other possible values of the physical parameters g and t. Since there are
no nondimensional parameters other than the unknown y/gt2, dimensional
analysis has almost provided the complete solution without any knowledge
of dynamics. Our choice of position and time as the dependent and independent parameters of the problem was entirely arbitrary. We might also
have considered, for example, the velocity of the body as a function of the
distance it has fallen.
2.2 Pendulum
Next, we consider the period T of a simple pendulum, under the usual
assumptions of a plane motion about a frictionless pivot in a vacuum. The
physical parameters of significance are the pendulum length l, mass m, and
maximum angle of its swinging motion θ0, together with the gravitational
acceleration g. A suitable multiplicative combination with the desired
units of time is (l/g)1/2, and the only nondimensional parameter that can be
formed from the four physical parameters is θ0, which is itself nondimensional. It follows that
T ( g / l )1/2 = f (θ 0 ),
(2)
where f is a nondimensional function of the maximum angle. If we also
assume that this angle is very small, and if we observe that under these
circumstances the period T tends to a finite limit, then
T ( g / l )1/2 = f ( 0 ).
(3)
The constant f(0) is known from mechanics to be 2π and could be measured
from a single experiment.
Rather than using the period T, we could employ the frequency 1/T or
the radian frequency ω = 2π/T. In terms of the latter parameter, the above
relations take the form
ω (l / g )1/2 = 2π / f (θ 0 ) 2π / f ( 0 ).
(4)
12
Chapter 2
Figure 2.1
Sketch of a periodic progressive wave in a fluid of mean depth h. Note that λ is the
wavelength, A the wave amplitude, and the wave translates with phase velocity Vp in
the direction shown by this vector.
2.3 Water Waves
As our first application of dimensional analysis to a problem in hydrodynamics, we shall consider the motion of an oscillatory wave system on the
free surface. In some respects this is a complicated problem with which
to commence our discussion of fluid motions, but the analogy with the
motion of a simple pendulum is useful. Moreover, the remaining problems
will involve the interactions of the fluid with a rigid body of some sort,
and it is convenient to discuss wave motions before inserting the body into
the fluid.
Figure 2.1 shows a periodic progressive wave system which is characterized by its amplitude A, wavelength λ, and period T. As in section 2.2,
the period T can be replaced by the radian frequency ω = 2π/T. The wave
motion propagates through the fluid with a phase velocity Vp such that any
prescribed point of constant phase, such as a crest or trough of the wave,
will progress a distance λ in time T. Thus
VP = λ / T = ωλ / 2π .
(5)
Like the simple pendulum, this wave motion is the result of a balance
between kinetic energy and potential energy. The most important physical
parameters are the fluid density ρ, gravity g, and the depth h. Thus, the
wave period T must be a function only of the wavelength λ, amplitude A,
density ρ, gravitational acceleration g, and depth h. Note that the frequency
ω and phase velocity Vp are not included here, since these are not independent and can be substituted for, or replaced by, T and λ.
Model Testing
13
As in the simpler problems of the falling mass and pendulum, only one
physical parameter, the density of the fluid, contains the units of mass.
Since it is impossible to nondimensionalize ρ in combination with the
other parameters, it can be deleted from the analysis. A total of five dimensional quantities (T, λ, A, g, h) remain, and a nondimensional form for the
period is
T ( g / λ )1/2 = f ( A / λ , h / λ ).
(6)
Comparing this result with equation (2), we find that the nondimensional
amplitude ratio A/λ is analogous to the maximum pendulum angle θ0, but
an additional depth parameter h/ λ must now be considered.
Two possible simplifications of (6) can be pursued. First, as in (3), if
the wave amplitude is sufficiently small compared to the wavelength, a
linearized result follows from (6),
T ( g / λ )1/2 f ( 0, h / λ ),
A / λ 1.
(7)
Alternatively, if the fluid depth is very large compared to the wavelength, as
in the deep ocean, (6) can be replaced by
T ( g / λ )1/2 f ( A / λ , ∞ ),
λ / h 1.
(8)
If both inequalities are satisfied simultaneously,
T ( g / λ )1/2 f ( 0, ∞ ) = constant.
(9)
Thus, for small amplitude waves in deep water, the period is proportional to
λl/2 and, from (5), the phase velocity likewise is proportional to the square
root of the wavelength.
As in the simpler problems of the falling mass and pendulum, the constant in (9) cannot be determined from dimensional analysis. A more complete solution of the linearized problem in chapter 6 will show that this
constant is equal to (2π)1/2 and will provide the more general functional
relation (7) for finite depth.
Since the phase velocity Vp depends upon the wavelength, it follows that
water waves of different wavelengths or periods will propagate at different
phase velocities and are thus dispersive. Long waves will travel faster than
short waves, and a spectrum of waves will constantly change its appearance in a manner obvious from observation of ocean waves. The monochromatic wave system shown in figure 2.1 is an exception, where the
14
Chapter 2
free-surface profile can propagate without change of form because only one
wavelength is present.
Other physical properties of secondary importance include the viscosity
of the water, the surface tension of the air-water interface, and the properties of the air. The fluid viscosity exerts a small dissipative effect, comparable to the friction of the pendulum bearing, and typical water waves
can travel for hundreds or thousands of wavelengths without significant
attenuation. Qualitatively this could be anticipated by forming a nondimensional viscosity ratio vg-1/2λ-3/2, which is equal to 3 × 10−4 for λ = 1 m and
is even smaller for longer wavelengths.
The surface tension of the air-water interface exerts a tensile force, per
unit width of the free surface, proportional to the curvature of the free surface and analogous to the restoring force of waves on a string or membrane.
A suitable nondimensional parameter is Ts/ρgλ2 where the surface tension
Ts for an air-water interface is about 0.07 N/m. Thus Ts/ρgλ2 is on the order
of 10−5 for λ = 1 m, and smaller for longer waves. It follows that surface tension effects are negligible for all but the shortest waves or ripples.
Generally one can ignore the effects of the air on water waves since the
air density is on the order of 10−3 times that of water. An important exception is the process by which ocean waves are generated by the wind, but
that is a complicated problem beyond the scope of this text.
2.4 Drag Force on a Sphere
Dimensional analysis is particularly useful in studying the forces exerted
by the fluid on a moving body. As our first example we shall consider the
drag force acting upon a sphere moving with constant velocity U through a
viscous fluid of unbounded extent. Of course, no fluid is truly unbounded,
and this restriction implies that the sphere is very small compared to the
distance to the nearest boundary or other body within the fluid.
If the surface of the sphere is smooth, the only length scale of the problem is the sphere diameter. The drag force D must be a unique function2 of
the diameter d, the sphere velocity U, the fluid density ρ, and the kinematic
viscosity coefficient υ. In the dimensional form
D = f ( d ,U , ρ , ν ).
(10)
Model Testing
15
Nondimensionalizing these five parameters yields two nondimensional
quantities, which can be expressed in the form
D
= f (Ud / ν ),
ρU 2 d 2
(11)
where R = Ud/υ is the Reynolds number based on the sphere diameter.
In this analysis the body weight W and buoyancy force ρ g∀, where
∀ = displaced volume, are not involved because we are considering only the
hydrodynamic drag force associated with steady translation of the body.
The translation may be in the horizontal plane or in the vertical direction
with the static weight and buoyancy forces subtracted from the total force.
On the other hand, if the body is freely falling (or rising), the excess weight
(W − ρ g ∀) is an important physical parameter. In the steady case, where
a freely falling body has reached its terminal velocity, the excess weight is
balanced by the steady-state drag force.
Writing (11) in a more conventional form, we have
1
2
D
= CD ( R),
ρU 2 S
(12)
where S = πd2/4 is the frontal area of the sphere and CD is the drag coefficient. Nothing more can be said about the drag, unless experimental observations of the drag coefficient are available for various Reynolds numbers,
but the information now required has been reduced to a bare minimum. To
predict the drag of any sphere in any fluid, it is sufficient to perform experiments with a single sphere over a range of velocities and in a single fluid.
This is illustrated in figure 2.2, which shows experimental measurements
of the drag coefficient for several fluids, including water and air, and for a
variety of sphere diameters.
An explanation of the results shown in figure 2.2 requires a description of fluid mechanics beyond the scope of dimensional analysis. Nevertheless, a brief qualitative discussion will provide an understanding of
this and other drag problems to be studied subsequently in this chapter.
For moderate Reynolds numbers, the dominant contribution to the drag
force is due to separation, which occurs near the midplane of the sphere.
Upstream of the separation point viscous effects are confined to a thin
boundary layer adjacent to the sphere surface, and the vicinity of the stagnation point at the front of the sphere will be a region of relatively high
Figure 2.2
The drag coefficient of a sphere (reproduced from Goldstein 1938). Note the transition from laminar to turbulent flow at a Reynolds number
of approximately 3 × 105. Dashed line for R < 10 is a low-Reynolds number theory due to Oseen.
16
Chapter 2
Model Testing
17
pressure compared to the free-stream value at infinity. Downstream, within
the separated wake, the fluid is relatively stagnant but has a reduced pressure dictated by the requirement of continuity with the free-stream pressure outside the separated region. Thus, as a result of separation, there is a
substantial pressure difference between the forebody and afterbody; from
dimensional considerations it must be on the order of magnitude of ρU2,
and the resulting drag coefficient is of order of magnitude unity. This situation ensues, in figure 2.2, over the range of Reynolds numbers between
103 and 3 × 105. In this regime the viscous flow in the boundary layer on
the forebody is laminar.
At a critical Reynolds number of about 3 × 105, the boundary-layer
flow becomes turbulent, and the resulting increase of momentum convection postpones separation. Thus, the separated region behind the sphere is
diminished, and the drag is reduced in a dramatic fashion that coincides
with transition of the upstream boundary-layer flow from the laminar to
turbulent regime. The precise point where transition to turbulence occurs,
with a resulting decrease in the drag of a bluff body, depends not only on
the body shape but also on the ambient turbulence of the flow and on the
roughness of the body. The indented surface of a golf ball is intended to
stimulate early transition and reduced drag.
2.5 Viscous Drag on a Flat Plate
As an example of a highly streamlined body shape, we consider a rectangular flat plate of length l, breadth B, and negligible thickness, moving with
velocity U in the longitudinal direction parallel to its length dimension.
The drag coefficient will depend not only on the Reynolds number but also
on the aspect ratio B/l of the plate. In general, we must anticipate a result
of the form
1
2
D
= CD ( R, B / l ).
ρSU 2
(13)
Here S is the surface area of the plate; we have chosen this in preference,
say, to B2 or l2 because we expect the viscous drag to be roughly proportional to the surface area over which viscous shear stresses act. This expectation is confirmed by the fact that, for Reynolds numbers on the order of 105
to 1010, the drag coefficient is relatively insensitive to the ratio B/l. This is
18
Chapter 2
not a foreseeable consequence of dimensional analysis; instead, it must be
explained from the characteristic thickness of the boundary layer.
Experimentally determined frictional-drag coefficients CF are shown in
figure 2.3 for various flat plates, along with the semiempirical equation
determined by Schoenherr:
0.242 / CF = log10 ( RCF ).
(14)
Here the coefficient has been changed from CD to CF because we must distinguish subsequently between total drag and frictional drag. Once again,
the validity of the dimensional analysis is confirmed by the collapse of data
from diverse experiments both in water and in air.
There is a noticeable scatter of data in the transition range of Reynolds
numbers between 105 and 2 × 106. In that range, the flow changes from a
smooth laminar regime to the turbulent regime that persists throughout
the range of higher Reynolds numbers. In this transition range an important mechanism that triggers turbulence is the smoothness of the body
surface. Thus, the drag of rough plates shifts to the turbulent value at lower
Reynolds numbers; for very smooth plates, laminar flow can be maintained
longer. Similar results occur if the ambient flow is not uniform.
The principal consequence of turbulence is to increase the momentum
defect of the boundary layer and the resulting frictional drag on the flat
plate. This is precisely opposite to the effect noted for a sphere where the
increase in frictional drag is insignificant compared with the decrease in
pressure drag. That the magnitude of the frictional drag coefficient shown
in figure 2.3 is two orders of magnitude less than that of the drag coefficient for a sphere emphasizes the practical importance of streamlined body
shape.
2.6 Viscous Drag on General Bodies
The drag on more general bodies can be determined from tests of geometrically similar models, or geosims, provided that the Reynolds number for
model and full-scale bodies is the same. However, practical difficulties may
prevent this scaling procedure from being carried out. Since it is difficult
to find nonvolatile liquids appreciably less viscous than water, the ratio of
model velocity to full-scale velocity must be inversely proportional to the
ratio of the lengths. As an example, let us consider a ship or submarine of
mental data see Todd (1967), figure 3.
Figure 2.3
Schoenherr’s flat-plate frictional drag coefficient, compared with various experimental results and the Blasius boundary-layer theory. Note
that transition occurs between R = 105 and 2 × 106, depending on the plate roughness and ambient turbulence. For references to the experi-
Model Testing
19
20
Chapter 2
100 m length, moving at 10 m/s or 20 knots; if a 10 m model of the same
vessel is to be tested in water at the same Reynolds number, it must move
with a velocity of 100 m/s or 200 knots.
This difficulty is usually overcome by separating the total viscous drag
into two components, frictional drag and pressure drag. These can be
defined as the longitudinal components of the forces acting on the body
due respectively to tangential shear stresses and normal pressure stresses.
We assume that the tangential stress is the only component affected by the
Reynolds number and, moreover, that the resulting frictional-drag force
is equal to that of a flat plate of equal area and Reynolds number. The
remaining force component, due to normal pressure stresses acting on the
body surface, will depend on the form of the body; for this reason it is
often called the form drag or pressure drag. In the case of a streamlined body
the form or pressure drag can be assumed independent of the Reynolds
number over the range where the boundary layer is thin compared to the
body dimensions, typically for R > 105. On the other hand, for a bluff
body the possible dependence of the point of separation on R must be
recognized.
With these assumptions, the total drag coefficient can be written in the
form
C D ( R ) = CF ( R ) + CP ,
(15)
where CF is the flat-plate frictional-drag coefficient defined in the last
section, and plotted for a wide range of Reynolds numbers in figure 2.3. The
pressure-drag coefficient Cp can be determined, for the body in question,
by performing a model test at a convenient Reynolds number and by using
data such as that in figure 2.3 to predict the corresponding value of CF.
For streamlined three-dimensional bodies with maximum thickness less
than about one fifth of the length, the frictional drag is dominant and
the drag coefficient can be predicted from the flat-plate drag coefficient
CF. The pressure drag is more important for two-dimensional cylinders or
struts, because of the increased obstruction of the flow past the body in two
dimensions. Figure 2.4 shows the total drag coefficient for a circular cylinder, two streamlined struts with thickness-length ratios of 1/2 and 1/4, and
the limiting case of zero thickness given by twice the flat-plate frictional
drag coefficient. This figure illustrates the conflicting roles of transition to
Model Testing
21
Figure 2.4
Drag coefficient as a function of Reynolds number for two-dimensional cylinders
of maximum thickness-to-length ratio t/l. For the flat plate (t/l = 0) the curve shown
is twice the skin-friction coefficient of figure 2.3, and the shaded areas of this and
the circular cylinder indicate the general range of uncertainty depending on ambient turbulence and roughness of the surface. For the intermediate cylinders, the
precise values of the drag coefficients will depend on the shape, especially near
transition.
turbulence, which decreases the drag of bluff bodies and increases the drag
of fine bodies.
If the body has sharp edges not aligned with the flow, separation will
generally occur at these edges irrespective of the Reynolds number or the
state of the upstream boundary layer. For example, a circular disc moving normal to its plane will experience separation at the periphery; the
drag coefficient for this body is about 1.1 for all Reynolds numbers greater
than 103.
In summary, the drag coefficient of a body moving in a viscous fluid will
depend in general on the Reynolds number; but for a bluff body at large
Reynolds numbers the drag coefficient is insensitive to Reynolds number
except insofar as this affects the point of separation.
22
Chapter 2
2.7 Hydrofoil Lift and Drag
Hydrofoils, rudders, propeller blades, yacht sails, and keels are diverse
examples of lifting surfaces. Generally these are thin streamlined bodies,
intended to develop a hydrodynamic lift force L. The term hydrofoil will be
used in a generalized sense to encompass all possible applications of lifting
surfaces, as opposed to the strict definition of the supporting devices for a
hydrofoil boat.
The performance of hydrofoils in water is analogous to that of airplane
wings (and sails) in air, the principal distinction between these being the
density of the fluid medium. The term lift derives from the aeronautical
context and is used for the corresponding force component on a lifting
surface regardless of its orientation in space. In particular, the lift force on a
rudder, yacht sail, or keel is generally horizontal.
Inevitably, hydrofoils experience a drag force D, and the total force F is
a vector quantity, with components (D, L). In a reference frame moving
with the hydrofoil, the drag component is in the same direction as the free
stream, as shown in figure 2.5. By definition, the lift force is perpendicular
to the free stream, regardless of the angle of attack, α, between the orientation of the hydrofoil and the free-stream direction.
Figure 2.5
Flow past a hydrofoil section. The angle of attack α is generally defined with respect
to the “nose-tail line,” between the center of the minimum radius of curvature of
the leading edge and the sharp trailing edge. L and D denote the lift and drag components of the total force F, and are defined respectively to be perpendicular and
parallel to the free-stream velocity vector.
Model Testing
23
In this section the dimensional analysis of lifting surfaces is outlined in
a form that can be used with model tests to predict the lift and drag forces
acting on full-scale hydrofoils. Geometrical similarity is assumed, as in the
earlier sections of this chapter.
For the two-dimensional case shown in figure 2.5, the length scale of
the hydrofoil or its model can be described by the chord length l. In three
dimensions, the lateral extent of the hydrofoil is its span s. For geometrically similar hydrofoils, either l or s can be used as the characteristic
length for dimensional analysis of the lift and drag force. However, as in
the case of the viscous drag on a flat plate, the results will be more general
if we anticipate that the force acting on a lifting surface is approximately
proportional to the planform area S. For a planar hydrofoil this can be
defined as the projected area of the foil in the direction of the lift force,
with α = 0.
We shall assume that the hydrofoil operates in a steady-state manner
with constant velocity and angle of attack, in an unbounded fluid, and
that the ambient pressure is sufficiently high to preclude cavitation. Thus
the relevant physical parameters affecting the hydrodynamic forces are the
planform area S, velocity U, angle of attack α, fluid density ρ and the kinematic viscosity υ). The lift and drag forces can be expressed in the nondimensional forms
1
2
L
= CL ( R,α ),
ρU 2 S
(16)
1
2
D
= CD ( R,α ).
ρU 2 S
(17)
As in the case of the flat plate, viscous effects depend primarily on the
length scale of the body parallel to the free-stream direction. For this reason
the chord length l is used to define the Reynolds number R = Ul/υ.
Values of the lift and drag coefficients can be obtained experimentally
by placing a scale model in a wind tunnel or water tunnel. The angle of
attack can be varied by a suitable arrangement and the force components
measured with dynamometers. Here again, however, there is a practical
difficulty in preserving the full-scale Reynolds number. Thus the drag is
generally treated as for a streamlined nonlifting body, accounting for the
Reynolds number dependence simply in terms of the frictional drag of a
24
Chapter 2
flat plate at zero angle of attack. The remaining pressure drag is assumed to
depend only on the angle of attack, and therefore
CD ( R,α ) CF ( R) + CP (α ).
(18)
Usually the lift coefficient is assumed to depend only on the angle of attack,
and to be independent of the Reynolds number, in the form
CL ( R,α ) CL (α ).
(19)
In practice, many hydrofoils are of large span, relative to their chord.
Under these circumstances the flow at each section along the span is
approximately two-dimensional, and the total three-dimensional force
coefficients can be estimated by integration of the sectional lift and drag
coefficients along the span. The differential projected area dS is proportional to the product of the local chord length and the differential element
of the distance along the span. Thus the two-dimensional sectional lift and
drag coefficients nondimensionalized in terms of the local chord lengths
are consistent with the definition of the total force coefficients based on
the planform area S.
Typical experimental results for the sectional lift and drag coefficients
are reproduced in figures 2.6 and 2.7, for the foil geometry shown in figure 2.5. The validity of the approximations (18–19), over a limited range
of Reynolds numbers, can be judged from figures 2.6 and 2.7. For small
angles of attack the lift coefficient shown in figure 2.6 is insensitive to the
Reynolds number, and CL increases in a linear manner with the angle of
attack. As the angle of attack increases, however, the streamlined effect of
the foil diminishes, and ultimately separation, or stall, occurs with a dramatic reduction in the lift coefficient. The point at which stall takes place
is sensitive to the Reynolds number, as well as to the ambient turbulence of
the fluid and to the roughness of the foil surface; this situation is analogous
to the drag coefficient for a sphere or other bluff body.
Figure 2.7 shows the drag coefficient. For small positive or negative
angles of attack where the lift force is small, the drag coefficient is insensitive to the Reynolds number and takes a value comparable to the flat plate
frictional drag coefficient. As stall is approached, the drag goes up appreciably and in a manner sensitive to the Reynolds number. Since the magnitude
of the drag coefficient is small by comparison to the lift, these effects are of
less importance.
Model Testing
25
Figure 2.6
Lift coefficient (16) for a two-dimensional hydrofoil. The results here are for the
NACA 63–412 section which is shown in figure 2.5. The dashed curve (— — —)
shows the behavior at stall for a foil with artificial roughness near the leading edge,
for R = 6 × 106. (Adapted from Abbott and von Doenhoff 1959)
2.8 Screw Propeller
A screw propeller consists of several hydrofoil-type lifting surfaces, arranged
in a helicoidal fashion to produce lift and thrust when rotated about the
axis. The dimensional analysis of propellers is similar to that of the simpler
hydrofoil, except for a change in the conventional set of physical parameters. We now consider the required shaft torque Q and resulting axial thrust
force T associated with a propeller of diameter d, which rotates with constant angular velocity n while simultaneously moving in the forward (axial)
direction with velocity U, as shown in figure 2.8. Note that n is the number
of shaft revolutions per unit time; thus, the peripheral velocity of the blade
at a radius r is 2πnr. A velocity diagram of the blade element at r is as shown
26
Chapter 2
Figure 2.7
Drag coefficient (17) for the NACA 63–412 section as described in figure 2.6. (Adapted from Abbott and von Doenhoff 1959)
in figure 2.9, which may be compared with figure 2.5. The angle of attack is
measured now by the nondimensional advance ratio J = U/nd.
The results of section 2.7 suggest that Reynolds-number effects will be
negligible, provided the propeller is not operated in a regime of excessive angle of attack. Similarly, we assume no cavitation. It follows that the
thrust and torque can be nondimensionalized so as to depend only on the
advance ratio, in the form
T
= KT ( J ),
ρn 2 d 4
(20)
Q
= KQ ( J ).
ρn 2 d 5
(21)
The propeller efficiency ηp is the ratio of the work done by the propeller in
developing a thrust force UT, divided by the work required to overcome the
shaft torque 2πnQ. Thus, it follows that
Model Testing
27
Figure 2.8
Perspective sketch of a propeller and its shaft.
Figure 2.9
Two-dimensional view of a propeller blade section, moving to the right with velocity U and in the peripheral direction with velocity 2πnr. The local angle of attack α
is the difference between the inflow angle tan−1 (U/2πnr) and the pitch angle of the
blade. The resultant force F contains an axial component (thrust), and a peripheral
component (torque/radius).
28
Chapter 2
ηP =
UT
J KT
.
=
2π nQ 2π KQ
(22)
Figure 2.10 shows values of the thrust and torque coefficients and the
efficiency for a series of propellers of varying pitch, which is the axial distance of advance in one complete revolution along the helix of the blades.
The thrust and torque coefficients are decreasing functions of the advance
ratio J, and the thrust coefficient vanishes at a value of J such that the propeller blades are at zero angle of attack with respect to the incoming fluid.
At this same condition there is still a small residual torque, due to frictional
drag on the blades and hub. At a slightly higher value of J, the propeller
“windmills,” requiring no torque but with a small drag force. For propeller blades of pure helicoidal shape, the thrust would vanish in the absence
of viscous friction at the point where the advance ratio is precisely equal
to the pitch-diameter ratio. However, the blades have some camber and,
therefore, develop a small amount of lift and thrust at zero geometrical
angle of attack.
Decreasing the advance ratio J increases the angle of attack, as shown in
figure 2.9. Thus, in figure 2.10, the thrust and torque coefficients increase
monotonically with decreasing values of J. At J = 0 the propeller operates
in a regime of large thrust and torque, corresponding to the bollard-pull
test of a tug boat exerting a force on a fixed object. The efficiency of this
regime is zero, as defined by (22), and the results of figure 2.10 show that
the optimum efficiency of a propeller is achieved at a relatively small angle
of attack, with maximum efficiencies on the order of 0.6 to 0.8.
If cavitation occurs, the thrust coefficient will decrease due to the corresponding loss of lift of each blade section. Under these circumstances the
torque is also reduced, but not to the same extent, and thus the propeller
efficiency decreases. Supercavitating propellers, designed to operate efficiently in the cavitating regime, have been developed for use in high-speed
vessels. As a rule, these propellers are somewhat less efficient than a conventional subcavitating propeller but more efficient than a conventional
propeller that is cavitating. Supercavitating propellers are discussed by Venning and Haberman (1962) and Todd (1967). Charts for the characteristics of conventional propellers, operating in both regimes are given by van
Lammeren, Troost, and Konig (1948), and extended by van Lammeren, van
Manen, and Oosterveld (1969).
Model Testing
29
Figure 2.10
Thrust, torque, and efficiency coefficients for a series of three-bladed propellers
with pitch/diameter ratios 0.6, 0.8, 1.0, 1.2, and 1.4. (Adapted from van Lammeren,
Troost, and Konig 1948)
2.9 Drag on a Ship Hull
Next we consider a ship hull moving with constant velocity U on the free
surface. As in the previous examples, geometrical similarity is assumed
with the geometry of the hull characterized by its length l. Since the steady
motion of a ship on the free surface generates a system of waves dependent
on the gravitational acceleration g, this parameter must be included in the
dimensional analysis along with the drag D, length l, velocity U, density ρ,
and kinematic viscosity υ. These six dimensional quantities can be reduced
to three nondimensional ratios; the usual form for the drag coefficient is
given by
1
2
D
= CD ( R, F ).
ρSU 2
(23)
Here S is the wetted-surface area of the hull, and the Froude number
F = U/(gl)1/2 represents the effect of gravity.
30
Chapter 2
Aside from the practical difficulty of testing small models at the full­
scale Reynolds number, it is impossible now to scale simultaneously both
the Reynolds and Froude numbers, as noted in chapter 1. Therefore, one
cannot determine the drag of a ship hull from a suitable experiment with
a small-scale model, unless additional assumptions are made. We assume
that the drag coefficient (23) can be expressed as the sum of a frictionaldrag coefficient, depending on the Reynolds number, plus a residual-drag
coefficient which now must depend on the Froude number. Thus, we adopt
Froude’s hypothesis in the form
CD ( R, F ) CF ( R) + CR ( F ).
(24)
Here CF is the flat-plate frictional-drag coefficient, and CR is the residualdrag coefficient. Without such an assumption, one can express CD as the
sum of a flat-plate frictional-drag coefficient CF, which is a function only of
the Reynolds number, plus a residual-drag coefficient CR defined as
CR ( R, F ) = CD ( R, F ) − CF ( R),
(25)
but it is an approximation to assume that the resulting CR is independent
of the Reynolds number.
Before testing Froude’s hypothesis experimentally, one might consider
the physical motivations for this decomposition of the total drag. In section 2.6 a similar assumption was made; the additional component was a
constant independent of Reynolds number and associated with the form
drag or normal pressure force. Here the pressure force is augmented significantly by the drag associated with the formation of waves on the free
surface. Work must be done by the ship hull on the surrounding fluid to
generate the waves, and there is an associated drag component known
as the wave resistance. Since the characteristics of the waves are governed
by gravity, the wave drag will depend on the Froude number; hence, the
residual drag is not a constant but a function of the additional nondimensional parameter F.
Part of the residual drag may be due to viscous form drag, which should
not depend on the Froude number; but since this part is a constant, it can
be added to the residual drag without violating the assumption that the
latter is independent of the Reynolds number. Moreover, as we shall see,
on all but the slowest of ships the dominant portion of the residual drag is
associated with wave resistance rather than viscous pressure forces.
Model Testing
31
An experimental validation of Froude’s hypothesis can be made by plotting CD vs. R, for different values of the hull length. Figure 2.11 shows results
of experimental measurements from geosim models ranging in length from
1.2 m to 9.1 m, together with the results from fullscale drag measurements
of the Lucy Ashton, a ship of 58 m length. For each model length experimental points are shown for several values of the Froude number, which is
identified here in terms of the full-scale ship’s speed in knots. According to
Froude’s hypothesis, all experimental points for the same Froude number
(represented in the graph by the same symbol) should be situated equal
distances above the flatplate frictional-resistance curve, labeled here as the
Schoenherr line, and Froude’s hypothesis can be evaluated according to the
degree to which this is true. Rather than attempt to evaluate the discrete
points, we can refer instead to the smoothed cross-curves labeled according
to the various ships’ speeds. These represent values of CD vs. R for constant
values of F. These curves are not strictly parallel to the Schoenherr line, but
it is clear from figure 2.11 that Froude’s method does succeed to a considerable extent in correlating the resistance of geosim ship hulls of widely
differing lengths.
The low-Froude-number asymptotic tendency in figure 2.11 is indicated
by coalescence of the results for full-scale speeds of 4 to 5 knots. At these
speeds the wave resistance is negligible, and the principal contribution
to the residual drag is viscous form drag. Thus the differences between
the flat-plate “Schoenherr” curve in figure 2.11 and the curve immediately above it can be attributed to viscous form drag, whereas the difference between the latter curve and the appropriate curve for each larger
Froude number is attributable to wave resistance. The most noticeable
departure from the Froude hypothesis in figure 2.11 is that the form drag
is not constant but increases with decreasing Reynolds number. This tendency has led to the suggestion that the form drag should be assumed
proportional to the frictional drag or, alternatively, that the frictional drag
curve should be more steeply sloped than the Schoenherr line. Two such
empirical alternatives, the “Hughes line” and the “I.T.T.C. line,” are shown
in figure 2.12. The I.T.T.C. (International Towing Tank Conference) line
has become the most widely accepted extrapolation to use with Froude’s
procedure.
For purposes of extrapolating model results to obtain full-scale resistance coefficients, adding a constant to the extrapolator will not affect the
Figure 2.11
Total drag coefficients of the Lucy Ashton and several geosim models of the same vessel (from Troost and Zakay 1954). The faired curves
represent constant values of the Froude number and, if Froude’s hypothesis were strictly valid, these would be parallel with spacing independent of the Reynolds number. Note that, even for this small full-scale vessel (58 m long), there is a large gap between the largest model
results and the full-scale results.
32
Chapter 2
Model Testing
33
Figure 2.12
Frictional drag coefficients or extrapolators. The ATTC and ITTC lines are those recommended by the American and International Towing Tank Conferences; the former is identical to the Schoenherr line. (From Todd 1967; reproduced by permission
of the Society of Naval Architects and Marine Engineers)
ultimate prediction of CD This fact can be emphasized by writing Froude’s
hypothesis in the form
(CD )ship = (CD )model − (CF )model + (CF )ship .
(26)
Therefore, the difference in level between the I.T.T.C. and Hughes line in
figure 2.12 is not significant to the resulting predictions based on the alternative employment of one curve or the other.
Further study of the geosim results in figure 2.11 shows that while the
viscous form-drag coefficient, as measured at low Froude numbers, increases
with decreasing Reynolds number, the drag coefficient curves for higher
Froude numbers display the opposite tendency. Thus, the residual-drag
coefficient actually decreases while the form drag is increasing. This is a
fortunate circumstance, since the corresponding errors in Froude’s hypothesis will cancel. However, it implies that both the wave drag and form drag
depend separately on Reynolds number. Fortunately these effects are small,
particularly for the larger models shown in figure 2.11, and it is apparent
34
Chapter 2
from this discussion why large towing tanks are built to accommodate
models of 7 m to 10 m length.
Finally, the use of a roughness allowance, which is generally taken to be an
additive contribution of +0.0004 to the full-scale value of CF, is attributed
to the significant surface roughness of full-scale hulls, due to rivet heads,
welding seams, marine fouling, and so forth. Such roughness effects are significant, and accounting for them in this way is not irrational if guided by
empirical results from full-scale trials. On the other hand, it is disconcerting
to find different values of the roughness coefficient used as a catchall to
account for various observed scale effects, based upon the performance of
ships; it is especially troublesome when the resulting value is negative. As a
recognition of this inconsistency, it is now common to refer to the correlation allowance, instead of to the roughness allowance.
2.10 Propeller-Hull Interactions
Having discussed propellers and ship hulls separately in sections 2.8 and
2.9, we should also comment briefly on their interactions. The most obvious of these is that the propeller operates in the wake of the hull; there the
velocity is generally retarded to a degree depending on the fullness of the
hull and the position of the propeller. A more subtle interaction results
when the suction effect of the propeller modifies the flow past the hull
and the resulting drag on the hull. Similar interactions may occur in other
contexts; in particular, when a second vessel operates in the wake of a body,
the characteristics of the flow in the wake are a principal determinant of the
second vessel’s performance.
The conventional experiment to study propeller-hull interactions is a
self-propelled test, which uses a model fitted with and propelled by a scalemodel propeller. The first difficulty with such a test is that with Froude
scaling, the drag coefficients of the model and full-scale vessel differ by
the Reynolds-number dependent difference in the respective frictionaldrag coefficients. Thus (CD)model > (CD)ship, and the extra thrust required to
propel the model must be furnished from the model propeller or from an
external force. To preserve the correct thrust and torque coefficients of the
model propeller, this extra differential thrust force is generally provided by
the towing carriage. However, this cannot account for all of the Reynoldsnumber dependent differences between the model and full-scale ship; in
Model Testing
35
particular, the boundary layer of the model will be thicker, and thus the
wake retardation of the model will be larger.
It is assumed that the model drag D is known from a towing test without the propeller and, similarly, that the open-water characteristics of the
propeller have been determined from a separate experiment or from charts,
such as figure 2.10. For the self-propelled test the propeller thrust T is prescribed from the balance of propulsion and drag forces, and the necessary
torque Q and shaft rotation rate n can be measured from the model.
The retardation of the wake behind the model is measured by the wake
fraction w. Thus, if U is the forward velocity of the hull through the undisturbed water and UA is the effective velocity of advance experienced by the
propeller in the presence of the hull, Then
w = 1 − U A / U.
(27)
Typically, 0 < w < 0.4; the larger value applies for very full hulls. The wake
fraction can be measured directly, with the propeller removed, or it can
be deduced indirectly from the thrust-identity condition, which yields the
Taylor wake wT. The Taylor wake is determined by substituting the thrust
coefficient KT, as determined from the self-propelled test, in the open-water
propeller chart, to determine the advance ratio J and the corresponding
advance velocity
U A = ndJ .
(28)
With this value of UA, the Taylor wake wT is defined by (27).
Since the wake behind the hull is not a parallel inflow with constant
uniform velocity, the correction of the propeller performance based simply
on a wake fraction is only approximate. While the thrust coefficients KT
are defined to be equal, there may be a small difference in the torque coefficients KQ, as shown in figure 2.13. This difference is accounted for by the
relative rotative efficiency
ηR =
( KQ )OW
,
( KQ )SP
(29)
where the subscripts OW and SP denote the open-water and self-propelled
conditions. In general, 1.0 < ηR < 1.1.
In the converse effect of the propeller on the hull, the suction of the
propeller generally reduces the pressure at the stern and hence increases
the drag force
36
Chapter 2
Figure 2.13
Relationship between self-propelled (SP) and open-water (OW) propeller tests, using
the thrust-identity condition to relate these two experiments.
D = (1 − t )T .
(30)
The coefficient t is the thrust-deduction coefficient, which is typically less than
0.2. In exceptional circumstances the thrust-deduction coefficient may be
negative. For example, the suction effect of the propeller may delay the
point of separation and reduce the resulting form drag. Another exception
is a supercavitating propeller, where the thickness of the cavities results in a
source-like flow and a positive pressure ahead of the propeller.
These interactions can be integrated to give an overall propulsive
efficiency. The power input is the product of the propeller torque and rotation rate, while the useful work done is the product of the drag of the hull
and its velocity. The propulsive efficiency is thus given by
η=
DU
(1 − t )UKT
(1 − t )
=
=
ηPηR ,
2π nQ 2π nd( KQ )SP (1 − wT )
(31)
where ηp is the open-water propeller efficiency, as defined by (22). The ratio
(1 − t)/(1 − wT) is known as the hull efficiency ηH; as a rule it takes a value
between 1.0 and 1.2. Thus the propulsive efficiency is generally greater
than the open-water propeller efficiency.
Model Testing
37
2.11 Unsteady Force on an Accelerating Body
If the relative motion between a body and the surrounding fluid is unsteady,
the hydrodynamic force F(t) exerted on the body will vary with time. To
simplify this problem as much as possible, we consider first the special
case of a body accelerated impulsively from a state of rest, at time t = 0, with
a constant velocity U thereafter. The fluid is assumed to be unbounded, as
in sections 2.4 to 2.6 where the corresponding steady-state problem was
discussed.
The time t may be nondimensionalized in the form Ut/l, the number of
body lengths traveled in a time t. We nondimensionalize the force as before
and anticipate that this will act predominantly in the direction opposite to
the body motion; then
1
2
−F
= CF ( R, Ut / l ).
ρU 2l 2
(32)
Here R denotes the Reynolds number Ul/υ, and F is the force acting in the
same direction as the body motion.
From the standpoint of dimensional analysis, equation (32) cannot
be simplified further, and experiments must be conducted with the two
parameters R and Ut/l varied independently to determine the nondimensional force coefficient CF. Such experimental information is sparse and
depends on the details of the initial acceleration. Thus, we focus our attention on the two limiting cases of large and small values of time. Results
from these cases will add considerable physical information and enable us
to predict the unsteady force for more general body motions with realistic
acceleration histories.
For large values of the nondimensional time Ut/l, the force coefficient
(32) should approach the steady-state drag coefficient. Thus
CF ( R, Ut / l ) CD ( R),
Ut / l 1.
(33)
The complementary limit of small time is less obvious but of comparable
importance. During the initial stage of rapid acceleration of the body and
surrounding fluid, inertial effects will dominate the viscous stresses in the
fluid, and (32) can be approximated by the inviscid limit
CF ( R, Ut / l ) CF ( ∞,Ut / l ),
Ut / l 1.
(34)
38
Chapter 2
Considerable insight into this unsteady flow problem can be gained
from the classical photographs made by Prandtl and his coworkers (figure
2.14). These show the flow past a circular cylinder accelerated impulsively
to a constant velocity. The first of these photographs, taken shortly after
the motion commences, confirms that the flow past the cylinder is a symmetrical attached flow; it is essentially identical to the flow to be analyzed
in chapter 4 on the assumption that the fluid is ideal with vanishing viscosity. For subsequent times shown in figure 2.14, the effects of viscosity
Figure 2.14
Initial stages of the flow past a circular cylinder which is accelerated impulsively from
a state of rest to constant velocity. (From Prandtl 1927)
Model Testing
39
increase; the boundary layer along the cylinder surface grows and, more
importantly, the wake separates downstream. Initially the separated wake
consists of two symmetric vortices, but with increasing time these vortices
grow in strength. Ultimately this arrangement becomes unstable; the vortices are shed alternately from the body and convected downstream in a
staggered configuration. The oscillatory nature of this flow for large time
contradicts the assumption of a steady state, but for practical purposes the
drag force approaches an essentially constant limiting value, corresponding
to the steady-state drag coefficient (33).
The force coefficient (34), which applies for small time t, is the result of
accelerating the body in an ideal fluid. As will be confirmed in chapter 4,
this force is simply proportional to the acceleration of the body, in manner
analogous to Newton’s equation F = ma. The coefficient of proportionality is the added mass or the effective mass of the fluid that surrounds the
body and must be accelerated with it. We will denote the added mass by
m11. Here the letter refers to the analogy with the body mass m in Newton’s equation, and the subscripts denote the directions of the force and
body motion. During the initial stage where viscous forces are negligible, it
follows that
F = − m11U ,
(35)
where U denotes the acceleration, and the force coefficient (34) takes the
form
CF ( ∞,Ut / l ) =
1
2
m11U
.
ρU 2l 2
(36)
The relative magnitudes of the viscous-drag force and the added-mass
force are measured by forming the ratio of the coefficients (33) and (36).
Since the added mass is proportional to ρl3, it follows that
CF ( R, ∞ )
=
CF ( ∞,Ut / l )
1
2
ρU 2l 2CD ( R)
.
∝ U 2 / Ul
m11U
(37)
For impulsive acceleration from a state of rest to a constant velocity U, the
is small during the period of acceleration and large subseparameter U 2 / Ul
quently. Thus added-mass forces will dominate initially, and viscous forces
subsequently, with a gradual transition between these regimes as shown in
figure 2.15.
40
Chapter 2
Figure 2.15
Force coefficient for a circular cylinder accelerated from an initial state of rest to a
constant velocity U, based on experiments of Sarpkaya (1966).
The results in figure 2.15 are for a circular cylinder, synthesized from
experiments of Sarpkaya (1966) in which the acceleration is practically constant for a short period of time and the velocity is constant thereafter. The
Reynolds numbers for this experiment are R ≃ 105. The details of the initial
force peak depend on the magnitude and duration of the acceleration.
2.12 Vortex Shedding
When a bluff cylindrical body such as a circular cylinder is moved with
constant velocity normal to its axis, one might expect the resulting flow to
be steady and laterally symmetrical. In fact, neither of these assumptions is
correct, because of the oscillatory shedding of vortices into the wake shown
in the last photograph of figure 2.14.
The ultimate configuration of vortices downstream in the wake is antisymmetrical as predicted from a stability analysis by von Kármán, and the
staggered array of shed vortices is known as a Kármán street. The vortices
are shed into the wake at a frequency f = ω/2π, and as a result of their antisymmetrical arrangement a lateral lift force will act on the body with the
same frequency. The magnitude of the lift force and the frequency can be
nondimensionalized in the forms
Lmax / 21 ρU 2l = CL ( R),
(38)
fl / U = S( R),
(39)
Model Testing
41
where Lmax denotes the maximum value of the lift force, per unit length
along the cylinder, and S is the Strouhal number. These parameters have
been measured experimentally, especially for circular cylinders where CL
is typically equal to 0.5 and the Strouhal number is about 0.22 for Reynolds numbers in the laminar regime. As noted by Roshko (1961), the Strouhal number for circular cylinders appears to be proportional to the inverse
of the drag coefficient. For Reynolds numbers between 102 and 107, but
excluding the transitional regime R ≃ 106, the Strouhal number is approximated closely by S = 0.23/CD.
From an engineering standpoint, the oscillatory lift force may be of more
practical importance than the drag, especially if a hydroelastic resonance
occurs between the Strouhal frequency and a structural mode of vibration
of the body. Strumming oscillations of cables are a consequence of this
phenomena, as well as lateral vibrations induced upon pilings and other
fixed structures in a current. The magnitude of the oscillatory lift force can
be reduced by cable fairing or simply by installing a small splitter plate on
the after-side of the cylinder. For long slender cylinders with axis normal to
the flow direction, the correlation of the vortices along the cylinder axis is
particularly important, and small devices that destroy this correlation are
effective in reducing the total lift force. On the other hand, lateral vibrations of the body in response to the lift force will serve to correlate the
phase of the vortex shedding, thereby increasing the magnitude of the lift.
Extensive bibliographies of this subject are given by Mair and Maull (1971)
and Berger and Wille (1972). More recent work is described by Griffin and
Ramberg (1976).
2.13 Wave Force on a Stationary Body
Ocean waves are of particular interest to ocean engineers and naval architects because of the interactions between these waves and structures on
or beneath the free surface. Our first example of such a problem is the
unsteady force acting upon a fixed structure in the presence of waves. For
simplicity, we treat only the x-component of this force, but a similar analysis applies for the total force vector.
Assuming geometrical similarity, we can describe the structure by a
length scale l, which may be the length or the diameter of the body. If we
neglect surface tension effects and assume a plane progressive wave system
42
Chapter 2
similar to that described in section 2.3, then the magnitude of the unsteady
wave force can depend only on the density ρ, gravity g, viscosity υ, depth
h, body length l, and time t, as well as the wave amplitude A, wavelength λ,
and the angle of incidence β of the waves relative to the body axis. Thus, a
total of ten dimensional quantities must be related in the form
F = f ( ρ , g , ν , A, λ , β , h, l , t ).
(40)
Alternatively, seven nondimensional parameters can be defined to replace
(40); one choice is the force coefficient
F
= CF ( A / λ , h / λ , l / λ , β , R,ωt ).
ρ gl 3
(41)
Here we find it convenient to use the wave frequency ω, which is not an
additional independent parameter but one related to g, A, and λ by (6).
In this problem the Reynolds number R should be the ratio Ul/υ where
U is a typical velocity scale of the fluid relative to the body. Since the
oscillatory displacement of the fluid particles is proportional to the wave
amplitude A, the magnitude of the fluid velocity is ωA, and an appropriate
Reynolds number can be defined by
R = ω Al / ν.
(42)
In principle, there is no difficulty in conducting a model test with the
appropriate values of A/λ, h/λ, l/λ, and β. If the resulting wave force is measured over one or more cycles, all relevant values of ωt will be included.
However, the Reynolds number cannot be scaled properly, since it follows
from (6) and (42) that R ∝ l3/2. This situation is essentially analogous to the
steady drag force acting on a ship hull.
To overcome this dilemma, we need a simplifying assumption analogous
to Froude’s hypothesis for the drag on a ship hull. For this purpose we estimate the relative magnitudes of the viscous and inertial forces, as in (37).
Here, with U replaced by ωA and the fluid acceleration U by ω2A, it follows
that the ratio of viscous forces to inertial forces is proportional to
= A / l.
U 2 / Ul
(43)
First, we restrict our attention to the case of a large structure or a vessel
such as a ship hull where the ratio in (43) is small and viscous effects are
negligible. If we assume in addition, that the wave amplitude A is small
Model Testing
43
compared to the wavelength λ and depth h, then the force coefficient (41)
will be linearly proportional to A, with nonlinear contributions proportional to A2 neglected. The linearized fluid motion is sinusoidal in time,
with frequency ω, and (41) can be expressed in the form
CF = CFO cos(ωt + ε ),
(44)
where ε is a phase angle, CFO a force coefficient, and both depend on h/λ,
l/λ, and β but not on time. Alternatively, with U(t) the oscillatory velocity
of fluid relative to the body at some prescribed position such as the body
centroid, (44) can be replaced by the equivalent expression
CF = CMU + CdU .
(45)
CM and Cd are known respectively as the apparent mass and apparent damping
coefficients. These depend on l/λ, or on the wave frequency ω, and can be
determined experimentally by measuring the amplitude and phase of the
wave force acting on the body.
Since (45) derives from (44), U(t) must be harmonic in time despite
the appearance of (45). However, a spectrum of waves with arbitrary time
dependence can be generated by superposition, and since (44–45) are linear, the resulting wave forces can be obtained by a similar process of superposition using these expressions.
The inertia and damping coefficients can be predicted theoretically. A
particularly simple limit occurs for a submerged body that is small compared with the wavelength, l/λ ≪ 1. Thus, from a theorem to be derived in
section 4.17, the force acting on the body is proportional to the local acceleration of the fluid relative to the body. Here, as opposed to the case of an
accelerating body in a fixed fluid, the constant of proportionality includes
an additional “buoyancy” force proportional to the displaced volume of
fluid ∀ and associated with the pressure gradient of the fluid in the absence
of the body. For l/λ ≪ 1 it follows that
CM ( m11 + ρ∀ ) / ρ gl 3 ,
(46)
Cd 0.
(47)
Thus, in the special case where A ≪ l ≪ λ, the wave force is given by the
approximation
F ( m11 + ρ∀ )U .
(48)
44
Chapter 2
When the body is small compared to the wave amplitude, however,
the situation is fundamentally different. From (43) and (37) viscous drag
forces will be dominant, and an obvious representation of the wave force is
given by
F = 21 ρl 2U |U |CD ( R).
(49)
In this expression the square of the velocity U has been replaced by the
product U|U| to ensure that the viscous drag force (49) acts in the same
direction as the fluid velocity. The utility of the approximation (49) is
diminished because the Reynolds number is itself a function of the instantaneous velocity. However, (49) is relatively useful for bluff bodies, where
the viscous drag coefficient is not sensitive to the Reynolds number.
The intermediate case, where A/l is a quantity of order one, is one of the
most important and least understood problems of this field. It is important because many structures, such as the risers of offshore platforms and
other cylindrical pilings, have diameters of about the same magnitude as
the typical wave amplitude. In this regime viscous and inertial effects are
of comparable magnitude, and one must return to the exact force coefficient (41). However, the inherent interactions between viscous and inertial
effects have not prevented engineers from simplifying expressions for the
wave force. The most common approximation is Morison’s formula, which
assumes that the total wave force is the sum of the inertial force (48) and
the viscous force (49),
F = ( m11 + ρ∀ )U + 21 ρl 2CDU |U |.
(50)
Since the validity of each term in (50) is restricted to a regime where A/l is
respectively small or large, the justification for Morison’s formula is strictly
pragmatic and must rest with experimental confirmation.
Considerable experimental effort has been devoted to validating Morison’s formula (50). For submerged bodies an approximation similar to (50)
appears valid for engineering purposes, provided the coefficients are determined experimentally and for appropriate values of the Reynolds number
and the parameter A/l. For bodies such as vertical surface-piercing pilings,
no satisfactory experimental confirmation of (50) exists; the reasons for this
are not understood adequately. Surveys of this subject have been made by
Hogben (1974) and Milgram (1976).
Model Testing
45
2.14 Body Motions in Waves
If ocean waves are incident upon a freely floating body, the unsteady wave
force discussed in section 2.13 causes the body to oscillate. The body motion
is often more important than the components of the hydrodynamic force,
although a study of the forces is essential to the theoretical prediction of
the body motions.
In this section we focus our attention on the vertical heave oscillations
of an unrestrained body in waves. The body may be floating on the surface,
or it may be submerged beneath the surface; but in both cases, the unrestrained body is neutrally buoyant, and the body mass m is equal to the
displaced mass of water ρ∀ .
If coupling effects are neglected, the heave motion y(t) results from equilibrium between the vertical hydrodynamic force and the product of the
body mass multiplied by the vertical acceleration ÿ. Thus, the body mass m
must be added to the nine physical parameters upon which the force (40)
depends. In nondimensional form, it follows that
y / A = f ( A / λ , h / λ , l / λ , β , R,ωt , m / pl 3 ).
(51)
Since m = ρ∀, the last parameter in (51) is equal to ∀ / l 3. For a prescribed
body shape this parameter is independent of the length scale; thus the last
parameter in (51) may be deleted in our study of unrestrained bodies.
The dependence of (51) on the Reynolds number is once again a source
of fundamental problems for model testing, as well as for making theoretical predictions of body motions. For large bodies, where A/l is small,
it follows from (43) that viscous effects are negligible. Moreover, for small
bodies where A/l is of order one, or large compared to one, the situation
is considerably simpler than that discussed in conjunction with the wave
force on a fixed structure. Here, a small unrestrained body will bob up and
down like a cork upon the sea, with little or no relative motion between
itself and the surrounding fluid. Thus, there will be no significant viscous
drag for small unrestrained bodies in waves, and viscosity may be neglected
in general.
Subject to these assumptions, (51) reduces to the simpler form
y / A = f ( A / λ , h / λ , l / λ , β ,ωt ).
(52)
46
Chapter 2
If the wave amplitude is small compared to the wavelength and depth,
nonlinear effects can be neglected. The linearized approximation to (52)
then takes the form
y / A = f ( 0, h / λ , l / λ , β ,ωt ) = f 0 ( h / λ , l / λ , β )cos(ωt + ε ),
(53)
where for a prescribed body shape, the amplitude f0 and phase angle ε
depend only on h/λ, l/λ, and β. In deep water there is no dependence on
the depth ratio h/λ.
For axisymmetric bodies with vertical axis, the heave response is independent of the angle of incidence β. As a particular example, figure 2.16
shows the heave-amplitude ratio for a slender spar buoy of draft T in deep
water. We note that f0 → 1 in the limit of T/λ ≪ 1, in accordance with our
presumption that a very small body will move with the same velocity as
the surrounding fluid. In the opposite limit, T/λ ≫ 1, the body is large
compared to the wavelength, and f0 → 0. At an intermediate wavelength of
about six times the draft, a sharp resonance occurs which is analogous to
the motion of a weakly-damped mechanical oscillator. This severe resonant
motion for the spar buoy is a consequence of vertical slenderness. We shall
see in the next section as well as in chapter 6 that most floating bodies
experience relatively moderate resonant motions by comparison with the
spar buoy shown in figure 2.16.
Two important exceptions occur where viscous forces significantly affect
the motions of unrestrained bodies in waves. First, if the body shape is such
that the inertial forces are small, then frictional forces due to viscous shear
will be important, as in the case of the steady drag on a flat plate or streamlined body. Second, cross-flow drag will be significant for a long slender
body if the length is such that the motions do not coincide with the local
wave velocity.
An obvious example of the first exception is the angular motion of a
body of revolution about its axis. Thus, the yaw moment on an axisymmetric buoy is due entirely to viscous effects. The rolling motion of ships
and submarines will be affected similarly if the body sections are nearly
circular. Viscous shear stresses are significant also in the heaving motions
of a slender spar buoy if the draft is very large compared to the diameter. In
the latter examples viscous damping is particularly important at resonance,
where the inertial and hydrostatic forces cancel each other.
Figure 2.16
Heave response of a slender spar buoy in regular waves, from Adee and Bai (1970).
The ordinate is the ratio of heave amplitude to wave amplitude, and the buoy is a
circular cylinder, with a conical bottom, as shown to scale in the sketch. The dashed
line is a theoretical prediction that neglects the hydrodynamic forces due to the motions of the body. The solid line includes a correction for the added mass. The circles
denote experimental measurements.
48
Chapter 2
The second exception occurs if the body is slender, with its length comparable to the wavelength and its transverse dimensions comparable to
the wave amplitude. If the body is rigid, it will move relative to the waves,
unlike a small body that is free to move with the local wave field. This situation is analogous to the case of a fixed body. Thus, the cross-flow drag force
can be estimated from (49), with U the relative lateral velocity component
between the body section and the wave field. This situation occurs for the
horizontal force acting on a slender spar buoy of large draft and for the vertical force on the hull of a semisubmerged stable platform. Cross-flow drag
is discussed in the latter context by Söding and Häusler (1976).
If the body is restrained by a mooring, the dynamics of this restraint
will affect the body motions in waves. In the simplest case the mooring
may be regarded as a linear elastic restraint and lumped with the hydrostatic restoring force acting on the body. For small bodies this restraint can
induce relative motion between the body and waves, and then viscous drag
forces must be included once again. If the mooring cable is very long, viscous forces on the cable are significant as well. The complexity of the latter
problem is aggravated by the difficulty of modeling the cable in a wave
tank of limited depth. A comprehensive account of moored buoys is given
by Berteaux (1976).
2.15 Ship Motions in Waves
As our final example, we consider the case of a ship hull moving with constant forward velocity U through a system of waves. In contrast with the
situation of section 2.14, this example adds one physical parameter U and a
corresponding nondimensional Froude number U/(gl)1/2. If the ship is unrestrained, if viscous effects are negligible, and if the wave amplitude is sufficiently small for linearization to be valid, then the heave-amplitude ratio
is a generalization of (53) in the form
y / A = f 0 ( h / λ , l / λ , β ,U / ( gl )1/2 )cos(ωt + ε ).
(54)
Similar nondimensional expressions hold for the other five degrees of
freedom.
Typical experimental results are shown in figure 2.17 for roll (angular
motion about the longitudinal axis) and pitch (about the lateral axis) of an
aircraft carrier in deep water. Here the Froude number is 0.23, corresponding
Model Testing
49
Figure 2.17
Roll and pitch response of a 319 m ship at 25 knots. The motions are nondimensionalized in terms of the maximum wave slope 2πA/λ. (From Wachnik and Zarnick
1965; reproduced by permission of the Society of Naval Architects and Marine
Engineers.)
to a full-scale speed of 25 knots. The roll amplitude is a maximum in beam
seas and, except for small experimental errors, is zero in head and following seas. For a given angle of incidence, the roll response is similar to the
buoy response shown in figure 2.16, although in some cases the resonance
is outside the range of frequencies shown. Similar comments apply to pitch,
except that the maxima now occur in head and following seas with zero
response in beam seas.
50
Chapter 2
Comprehensive discussions of this subject are given by KorvinKroukovsky (1961), Vossers (1962), and Lewis (1967). More recent contributions are included in Bishop and Price (1975).
Problems
1. In shallow water (h/λ ≪ 1), waves of small amplitude propagate with
a phase velocity independent of the wavelength. From a dimensional
analysis show that under these circumstances VP ∝ ( gh )1/2 and thus that
T ∝ λ ( gh )−1/2. Deduce the qualitative change in wavelength that will occur
as waves of constant period move toward a beach of gradually decreasing
depth.
2. Find the maximum density and weight of a spherical instrument package
of diameter 0.5 m whose terminal velocity does not exceed 0.3 m/s when
it is dropped in salt water at a constant temperature of 5°C. What are the
terminal velocities if this density is in error by ± 1%?
3. The drag of an oceanographic research submarine 9 m long is to be estimated from wind tunnel tests with a scale model of length 1.5 m and wetted surface area 1.2 m2. The measured drag force in the wind tunnel is 7.9
N at a wind speed of 30 m/s and air temperature of 25°C. If the full-scale
submarine is deeply submerged in salt water at 15°C, at what speed can its
drag be most accurately predicted? What is the best estimate of its drag at
a speed of 2 m/s? Can this estimate be refined by raising or lowering the
temperature in the wind tunnel?
4. A hydrofoil vessel weighing 100,000 N is supported by two identical
hydrofoils, with NACA 63–412 sections, set at a 3-degree angle of attack.
If the lift coefficients of these hydrofoils are as shown in figure 2.6 and the
chord length is 0.5 m, what must their span be to support the vessel at a
velocity of 20 knots?
5. A propeller is operating initially at J = 1.0. For the blade section at 70
percent of the maximum radius, find the value of the advance ratio where
the angle of attack is reduced by 5 degrees.
6. A motorboat with a drag of 8,000 N at the design speed of 6 m/s is
to be powered by an engine that can deliver 100 kw at a shaft speed of
1800 revolutions per minute. The maximum propeller diameter that can be
Model Testing
51
used is 0.6 m. (1) What is the optimum choice of propeller from the series
shown in figure 2.10 for operation at 1800 rpm? (Extrapolate if necessary.)
(2) What is the optimum if a reduction gear is used, and what is the resulting increase in propeller efficiency? (3) What percentage of the available
engine power is used in each case, assuming no mechanical losses in the
shaft or reduction gear?
7. A small motor vessel has the following drag at corresponding velocities: velocity, m/s: 2.0 3.0 4.0 5.0 6.0; drag force, N: 1200 2700 4800 9500
19,500.
An optimum propeller is to be chosen for this vessel with three blades,
diameter 0.6 m, and a fixed rotation rate of 10 revolutions per second. Plot
the required thrust coefficient as a function of the advance ratio J on the
same scale as figure 2.10. By observing the equilibrium points where this
thrust coefficient is equal to that of different pitch propellers, determine
the propeller pitch for maximum velocity. Discuss this choice from the
standpoint of propeller efficiency.
8. The total drag of a ship 200 m long, moving at 20 knots in salt water
(15°C), is to be determined from towing-tank tests of a 2 m model in fresh
water (15°C). The ship’s wetted surface is 6000 m2, and its displacement is
190 MN. (1) Find the weight of the model. (2) What is the wetted-surface
area of the model? (3) What speed should be used in the model test? (4) If
the model drag at this speed is 1.6 N, what is the predicted full-scale drag
based on the ITTC friction line? (5) What model speed would preserve the
full-scale Reynolds number?
9. For a hydrofoil submerged 1.0 m beneath the free surface, what is the
minimum velocity at which cavitation can be anticipated, if the cavitation
number for inception of cavitation is σ = 0.1. For a 1/10 scale model of
the same hydrofoil, operating at the same Froude number, find the atmopsheric pressure in a “vacuum towing tank” necessary to produce the same
cavitation number. Assume the temperature range 0–30°C, with the worst
possible temperature full scale and the best possible temperature in the
towing tank.
10. A model of a sailing yacht is towed with a small angle of attack, or
leeway angle, about the vertical axis. The drag force and side force are
measured separately. If the yacht hull operating in this manner can be
52
Chapter 2
regarded as a vertical hydrofoil in the presence of the free surface, what
are the appropriate scaling laws to apply to each force component? Distinguish between parameters that should be scaled in principle and those that
would be most important in a model test. In fresh water (15°C), towingtank tests of a model 2 m long with wetted-surface area 1.5 m2 give a
measured drag force of 10 N and a side force of 28 N at a speed of 1.4 m/s.
What are the corresponding full-scale results for a yacht 10 m long in salt
water (15°C)?
11. Construct a simple mathematical model for the propulsive equilibrium
of a canoe, assuming the canoe velocity Uc and paddle velocity Up are constants. If these velocities are both defined relative to the surrounding water,
show that the propulsive efficiency is
η=
  ( SCD )c 1/2 
Uc
= 1 + 

U c + U p   ( SCD ) p  
−1
where (SCD)c,p is the product of the area and the drag coefficient for the
canoe or paddle respectively. How does the efficiency depend on the size
of the paddle?
12. A submerged spherical body of volume 0.5 m3 and density equal to
half the density of water is moored in the deep ocean with a taut vertical
cable 1000 m long. Neglecting all hydrodynamic forces, but including the
upward buoyancy force on the body, find the tension in the cable and the
natural period of this inverted pendulum. Comment on how the unsteady
force in section 2.11 would affect this period. Estimate the horizontal excursion of the buoy and the angle of the mooring cable in a steady current of
0.5 knots.
13. Resonant strumming of a taut cable with circular cross-section is
observed at a flow velocity of 10 knots perpendicular to the cable. The
diameter of the cable is 2 cm. What is the natural frequency of the cable? Is
this situation likely to become more serious at higher speeds?
14. For an oscillatory flow U(t) = Aω sin ωt, normal to a circular cylinder of
diameter d, find the ratio A/d such that the viscous and inertial forces in
Morison’s equation (50) are equal. Assume that the added mass m11 is equal
to the displaced mass and that the drag coefficient based on the diameter
is equal to 1.0.
Model Testing
53
15. A floating spar buoy of draft 2 m contains an accelerometer to detect its
heave motion relative to an inertial reference frame, while floating freely in
waves. Based on the results shown in figure 2.16, predict the range of wavelengths for which this buoy can measure the wave height to an accuracy of
20 percent without making corrections for hydrodynamic effects.
References
Abbott, I. H., and A. E. von Doenhoff. 1959. Theory of wing sections. New York:
Dover.
Adee, B. H., and Bai, K. J. 1970. Experimental studies of the behavior of spar type stable
platforms in waves. College of Engineering, University of California, Berkeley report
NA-70–4.
Berger, E., and R. Wille. 1972. Periodic flow phenomena. Annual Review of Fluid
Mechanics 4:313–340.
Berteaux, H. O. 1976. Buoy engineering. New York: Wiley-Interscience.
Birkhoff, G. 1955. Hydrodynamics—a study in logic, fact, and similitude. New York:
Dover.
Bishop, R. E. D., and W. G. Price eds. 1975. International symposium on the dynamics
of marine vehicles and structures in waves. London: Institution of Mechanical
Engineers.
Goldstein, S., ed. 1938. Modern developments in fluid dynamics. 2 Vols. Oxford: Oxford
University Press. Reprinted 1965, New York: Dover.
Griffin, O. M., and S. E. Ramberg. 1976. Vortex shedding from a vibrating cylinder,
J. Fluid Mech. 75: 267–271 + 5 plates.
Hogben, N. 1974. Fluid loading of offshore structures, a state of art appraisal: Wave
loads. Royal Institution of Naval Architects Marine Technology Monograph.
Korvin-Kroukovsky, B. V. 1961. Theory of seakeeping. New York: Society of Naval
Architects and Marine Engineers.
Lewis, E. V. 1967. The motion of ships in waves. In Principles of naval architecture,
ed. J. P. Comstock, 607–717. New York: Society of Naval Architects and Marine
Engineers.
Mair, W. A., and D. J. Maull. 1971. Bluff bodies and vortex shedding—A report on
Euromech 17. Journal of Fluid Mechanics 45:209–224.
Milgram, J. H. 1976. Waves and wave forces. In Proceedings of the conference on the
behaviour of offshore structures (BOSS ’76). pp. 11–38. Trondheim: The Norwegian
Institute of Technology.
54
Chapter 2
Prandtl, L. 1927. The generation of vortices in fluids of small viscosity. Journal of the
Royal Aeronautical Society 31:720–741.
Roshko, A. 1961. Experiments on the flow past a circular cylinder at very high Reynolds number. Journal of Fluid Mechanics 10:345–356.
Sarpkaya, T. 1966. Separated flow about lifting bodies and impulsive flow about
cylinders. J. American Institution of Aeronautics and Astronautics 4:414–420.
Sedov, L. I. 1959. Similarity and dimensional methods in mechanics. New York:
Academic Press.
Söding, H., and F. U. Häusler. 1976. Minimization of vertical motions of floating
structures. In Proceedings of the conference on the behaviour of offshore structures (BOSS
’76). pp. 307–319. Trondheim: The Norwegian Institute of Technology.
Todd, F. H. 1967. Resistance and propulsion. In Principles of naval architecture, ed.
J. P. Comstock, 228–462. New York: Society of Naval Architects and Marine
Engineers.
Troost, L., and A. Zakay. 1954. A new evaluation of the Lucy Ashton ship model and
full scale tests. International Shipbuilding Progress 1:5–9.
van Lammeren, W. P. A., L. Troost, and J. G. Konig. 1948. Resistance, propulsion and
steering of ships. Haarlem, Netherlands: H. Starn.
van Lammeren, W. P. A., J. D. van Manen, and M. W. C. Oosterveld. 1969. The
Wageningen B-screw series. Society of Naval Architects and Marine Engineers Transactions 77:269–317.
Venning, E., and W. L. Haberman. 1962. Supercavitating propeller performance.
Society of Naval Architects and Marine Engineers Transactions 70:354–417.
Vossers, G. 1962. Behaviour of ships in waves. Haarlem, Netherlands: H. Stam.
Wachnik, Z. G., and E. E. Zarnick. 1965. Ship motion prediction in realistic shortcrested seas. Society of Naval Architects and Marine Engineers Transactions 73:100–134.
3 The Motion of a Viscous Fluid
Chapter
The
Motion
of
a
Viscous
3
Fluid
© Massachusetts Institute of TechnologyAll Rights Reserved
To develop analytical representations for the flow of a viscous fluid, it is
necessary first to describe the properties of the flow, especially the velocity
field, and the relevant forces that act on the fluid particles. Subsequently,
the physical laws expressing conservation of mass and momentum must
be invoked, in terms of this description, to derive the governing equations
of the flow. In addition, the role of viscosity must be described by a suitable hypothesis that relates the stresses between adjacent fluid particles to
the kinematic description of their relative motion. In this manner we shall
derive the equations of motion for a viscous fluid, as a system of coupled
nonlinear partial differential equations. These words may sound frightening to those not well trained in advanced calculus, and still greater apprehension will be felt by the opposite group who recognize that such a system
of equations generally does not possess simple solutions. Of course, most
fluid flows are not simple, so we must expect the governing equations to be
somewhat complicated.
Because of this complexity, we will restrict our discussion to very simple flows, particularly those involving infinitely long, flat, or cylindrical
boundary surfaces. These flows are of limited direct applicability, but they
will give us a qualitative “feel” for the role of viscosity and an important
quantitative tool in the form of boundary-layer theory. The importance of
viscosity for many problems of engineering interest is confined to a thin
layer adjacent to boundary surfaces. Furthermore, if the radii of curvature
of these surfaces are large compared to the boundary-layer thickness, the
flow will appear locally plane in the boundary layer.
There are exceptions to this convenient state of affairs; for example, in
low-Reynolds-number flows, the role of viscosity is not confined to a thin
boundary layer. This situation would hold for bodies of microscopic size, as
56
Chapter 3
in fluid suspensions or the swimming of microorganisms, but these are outside the usual scope of ocean engineering and naval architecture. A more
significant exception here is separation, for flows past bodies not suitably
streamlined. Separated flows are practically impossible to analyze. When
these exist, the boundary-layer theory is useful only for predicting the flow
upstream of the separation point and ultimately for predicting the occurrence of separation.
3.1 Description of the Flow
Adopting the Eulerian approach, we define a velocity vector V(x, y, z, t) to
be equal to the velocity of the fluid particle at the point x = (x, y, z) in a Cartesian, rectangular coordinate system at time t. In this coordinate system
the three components of V will be denoted by u, v, and w. Other physical
parameters that will be required are the fluid density ρ, which ultimately
will be assumed constant; the external force field F acting on the fluid particles, notably the gravitational force ρg; and the surface stresses, or forces
per unit area, which act upon adjacent surfaces of the fluid.
Each component of the surface stress must be defined not only by the
direction in which it acts but also by the orientation of the surface upon
which it is acting; hence, a total of 3 × 3 = 9 stress components must be
defined on a cubical surface aligned with the coordinate system, as shown
in figure 3.1. Thus, for example, the stress τyx acts in the y-direction upon a
surface of constant x.
This characterization of the stress components is actually too restrictive,
since it does not account for the more general situation where the orientation of the surface differs from one of the three principal planes of a Cartesian coordinate system. By a familiar argument that plays a similar role
in structural mechanics, however, the cube in figure 3.1 can be replaced by
an infinitesimal tetrahedron, with three orthogonal faces normal to the
Cartesian coordinates and one oblique face of arbitrary orientation. The
tetrahedron is assumed to be sufficiently small that the stresses are effectively constant along each face, and since the volume will be negligible
compared with the surface area, the surface forces will dominate the body
forces. Thus, the forces exerted on the four faces of the tetrahedron by the
surface stress components must balance. If n = (nx, ny, nz) denotes the unit
normal vector on the oblique face, each of its three components equals the
The Motion of a Viscous Fluid
57
Figure 3.1
Definition of Cartesian coordinates and stress tensor.
ratio of the area of the corresponding face to the area of the oblique face.
Thus, for the surface forces to be balanced, the component of the stress on
the oblique face, say in the x-direction, must be equal to τxxnx + τxyny + τxznz.
In other words, regardless of the angular orientation of a surface element,
the stress acting on it in a given direction may be decomposed into three
orthogonal components, as if it were a vector. Thus, the stress components
can be described in general by a second-order tensor.
A similar argument regarding the balance of moments can be used to
show that the stress tensor is symmetric. Thus, if the cube shown in figure
3.1 is sufficiently small so that surface moments dominate body moments
and changes in the magnitude of the stress components may be neglected
across the cube, then a positive value of τyx on the face shown will cause a
counterclockwise moment about the centroid. The moment on the opposite face will have the same sign, since the normal vector has the opposite
sense and hence a positive τyx acts downward. The only source of a balancing clockwise moment is the stress τxy acting in the x-direction on the top
and bottom faces. From this it follows that τyx = τxy; by similar reasoning,
τxz = τzx, τyz = τzy.
To avoid subsequent unwieldy algebra, we shall denote the Cartesian
coordinates by subscripts (1, 2, 3) with the convention that x = x1, y = x2,
and z = x3. Similarly, for the velocity components, u = u1, v = u2, and w = u3.
58
Chapter 3
Both notations will be used. The complete stress tensor may then be written
in any of the three following forms:
τ 11 τ 12 τ 13  τ xx τ xy

 
τ ij = τ 21 τ 22 τ 23  = τ yx τ yy
τ
 
 31 τ 32 τ 33  τ zx τ zy
τ xz 

τ yz  .
τ zz 
(1)
Since the stress tensor is symmetric, τij = τji.
Adopting a similar notation for the unit vector, n = (n1, n2, n3), the stress
component in the ith direction, on a surface element with unit normal n,
is given by the sum
3
∑τ
ij
n j ≡ τ ij n j ,
i = 1, 2, 3.
(2)
j =1
It is customary to delete the summation sign, as has been done in the last
form of this equation, with the summation convention: any term of an equation that contains the same index twice should be summed over that index.
It is also customary to assume that an equation such as this is valid for all
possible values of a free index, in this case i. Both these conventions are
employed hereafter, unless otherwise stated.
3.2 Conservation of Mass and Momentum
The conservation laws of physics can be related to a fluid provided we focus
our attention on a group of fluid particles or a material volume of fluid so
that we always examine the same group of particles. Thus we define a volume of fluid (t ) subject to the above restriction. If the fluid density is
denoted by ρ, the total mass of fluid in this volume is given by the integral
∫∫∫ ρ d . Conservation of mass requires that this integral be constant, or
d
ρ d  = 0.
dt ∫∫∫

(3)
Similarly, the momentum density of a fluid particle is equal to the vector ρV, with components ρui. Conservation of momentum requires that the
sum of all forces acting on the fluid volume be equal to the rate of change
of its momentum, with respect to a Newtonian frame of reference, or
d
ρui d  = ∫∫ τ ij nj dS + ∫∫∫ Fi d  .
dt ∫∫∫
S


(4)
The Motion of a Viscous Fluid
59
Here the surface integral is the ith component of the surface forces
acting on S, and the last volume integral is the sum of the body forces,
such as that due to gravity. The conventions stated at the end of section
3.1 apply.
Equation (4) can be put in a more convenient form, involving only volume integrals, by using the divergence theorem
∫∫∫ ∇ ⋅ Qd = ∫∫ Q ⋅ n dS.
(5a)
S

This can be rewritten in the indicial notation as
∂Q i
∫∫∫ ∂x

i
d  = ∫∫ Q i ni dS.
(5b)
S
Here Q is any vector that is continuous and differentiable in the volume
 , and the unit normal n is the exterior normal vector pointing out of  on
the surface S. Using (5b) to transform the surface integral in (4),
 ∂τ ij

d
ρui d  = ∫∫∫ 
+ Fi  d  .
∂
x
dt ∫∫∫
j


 
(6)
Equations (3) and (6) express the conservation laws of mass and momentum for the fluid, in terms of an arbitrary prescribed material volume (t ).
The need to consider this volume integral and especially its time derivative
is inconvenient, however. To overcome this problem, we first consider the
evaluation of the time derivative, bearing in mind that the volume of integration is itself a function of time.
3.3 The Transport Theorem
Let us consider a general volume integral of the form
I (t ) = ∫∫∫ f ( x, t ) d  .
(7)
 (t )
Here f is an arbitrary differentiable scalar function of position x and time t
to be integrated over a prescribed volume (t ), which may also vary with
time. Therefore the boundary surface S of this volume will change with
time, and its normal velocity is denoted by Un.
60
Chapter 3
In the usual manner of elementary calculus we consider the difference
∆ I = I (t + ∆t ) − I (t ) =
∫∫∫
 ( t + ∆t )
f ( x, t + ∆t )d  − ∫∫∫ f ( x, t )d  .
(8)
 (t )
Neglecting second-order differences proportional to (Δt)2, we have
f ( x, t + ∆t ) = f ( x, t ) + ∆t
∂f ( x , t )
.
∂t
A similar decomposition can be made for the volume (t ). Thus (t + ∆t )
differs from (t ) by a thin volume ∆ contained between the adjacent surfaces S(t + Δt) and S(t) and proportional to Δt. From equation (8) it follows
that
∆I =

∂f 
∫∫∫  f + ∆t ∂t  d − ∫∫∫ f d
 + ∆

∂f
= ∆t ∫∫∫
d  + ∫∫∫ f d  + O[( ∆t )2 ],
∂
t

∆
(9)
where the last term denotes a second-order error proportional to (Δt)2.
To evaluate the integral over the volume ∆, we note that this thin
region has a thickness equal to the distance between S(t) and S(t + Δt). This
thickness is the normal component of the distance traveled by S(t), in the
time Δt, which is equal to the product UnΔt. Thus the contribution from
the last integral is of first order, or proportional to Δt; to the same degree of
accuracy the integrand f may be assumed constant across the thin region in
the direction normal to S. Integrating in this direction only, we then have
∆ I = ∆t ∫∫∫

∂f
d  + ∫∫ (U n ∆t )f dS + O[( ∆t )2 ].
∂t
S
(10)
Finally, we obtain the desired result by dividing both sides by Δt and
taking the limit as this tends to zero; thus
dI
∂f
= ∫∫∫
d  + ∫∫ fU n dS.
∂t
dt
S

(11)
Equation (11) is known as the transport theorem. The surface integral in
this equation represents the transport of the quantity of f out of the volume
 , as a result of the movement of the boundary. In the special case where
S is fixed and Un = 0, equation (11) reduces to the simpler form where differentiation under the integral sign is justified.
The Motion of a Viscous Fluid
61
In another special case of particular interest,  is a material volume,
always composed of the same fluid particles; hence the surface S moves
with the same normal velocity as the fluid and Un = V∙ n = uini. In this case
it follows from (11) and the divergence theorem (5b) that
d
∂f
fd  = ∫∫∫
d  + ∫∫ fui ni dS
∫∫∫
dt  ( t )
∂t
S
= ∫∫∫
 (t )
{
}
∂f
∂
( fui ) d  .
+
∂t ∂xi
(12)
3.4 The Continuity Equation
Returning now to equation (3), expressing conservation of mass, we have
immediately from equation (12) that
d
∂
 ∂ρ

ρd  = ∫∫∫  +
( ρui ) d  = 0.
t
xi
dt ∫∫∫
∂
∂


 
(13)
Since the last integral is evaluated at a fixed instant of time, the distinction that  is a material volume is unnecessary at this stage. Moreover, this
volume can be composed of an arbitrary group of fluid particles; hence,
the integrand itself is equal to zero throughout the fluid. Thus, the volume
integration in equation (13) can be replaced by a partial differential equation expressing conservation of mass in the form
∂ρ
∂
( ρui ) = 0.
+
∂t ∂xi
(14)
Our intention from now on is to assume that the fluid is incompressible
and the density constant. Thus (14) can be simplified to give the continuity
equation
∂ui
= 0,
∂xi
(15a)
or, in vector form,
∇ ⋅ V = 0.
(15b)
62
Chapter 3
3.5 Euler’s Equations
If the transport theorem (12) is applied to the conservation of momentum
(6), it follows that,
∂
∂
∫∫∫  ∂t ( ρu ) + ∂x
i


 ∂τ ij

+ Fi  d  .
( ρui u j ) d  = ∫∫∫ 
∂
x


j
j


(16)
Once again the volume in question is arbitrary; hence equation (16) must
hold for the integrands alone, in the form
∂τ ij
∂
∂
( ρui ) +
( ρui u j ) =
+ Fi .
∂x j
∂x j
∂t
(17)
Finally, if the derivatives of products on the left side of this equation are
expanded by the chain rule, and conservation of mass is invoked from
equation (15), we obtain Euler’s equations in the form
∂ui
∂u
1 ∂τ ij 1
+ uj i =
+ Fi .
∂t
∂x j ρ ∂x j ρ
(18)
The left-hand side of Euler’s equations can be interpreted as the acceleration of a material particle of fluid, since the substantial derivative
D
∂
∂
∂
≡
+ V ⋅∇ =
+ uj
∂t
∂x j
Dt ∂t
(19)
expresses the time rate-of-change in a coordinate system moving with the
fluid particle.1
3.6 Stress Relations in a Newtonian Fluid
Finally, we must relate the stress tensor τij to the kinematic properties of the
fluid. The task here is analogous to specifying the stress-strain relations in
solid mechanics.
If the fluid is at rest, and more generally if there are no shear stresses, a
normal pressure stress will exist within the fluid. Equilibrium of the forces
acting across a small tetrahedron requires that this pressure be isotropic.
Thus, in the absence of viscous shear the stress tensor is
τ ij = − pδ ij ,
(20)
The Motion of a Viscous Fluid
63
where δij is the Kroenecker delta function, equal to 1 if i = j and 0 if i ≠ j.
By hypothesis, there are no viscous forces if the fluid moves as a rigid mass
without deformation, or with the velocity field
V = A + B × r.
(21)
Here A and B are constant vectors equal to the translation and rotation
velocities, and r is the position vector from the origin of rotation.
Viscous stresses will occur when the fluid velocity differs from (21), with
relative motion between adjacent fluid particles. The simplest example is a
uniform shear flow, shown in figure 3.2. Here, and for more general velocity fields, the fundamental assumption of a Newtonian fluid is that the stress
tensor is a linear function of the nine gradients ∂uk/∂xl. This ensures vanishing of the viscous stress components for uniform translation of the fluid.
The rotational term in (21) will be stress-free provided the gradients occur
only in the form of sums ∂uk/∂xl + ∂ul/∂xk. For an isotropic fluid, the values
of the stress components must be independent of the choice of coordinates,
and for flow in one plane there can be no shear stress in the direction
Figure 3.2
Stress and strain for simple shear flow of a Newtonian fluid.
64
Chapter 3
normal to this plane. The most general linear function of the velocity
gradients, consistent with these conditions and the requirement that τij is
symmetric, is of the form
τ ij = µ( ∂ui / ∂xj + ∂u j / ∂xi ), for i ≠ j.
(22)
The coefficient μ is the viscous shear coefficient, or simply the coefficient of
viscosity.
In order for τij to be a tensor, the only possible addition to (22) for
i = j is a second constant times the divergence ∂ui/∂xi, which vanishes for
an incompressible fluid. Thus, the total stress tensor for an incompressible
fluid is given by
 ∂u ∂u j 
τ ij = − pδ ij + µ  i +
,
 ∂xj ∂xi 
(23a)
or, in Cartesian components,
∂u

 2 ∂x

0
− p 0
 ∂ v ∂u


{τ ij } =  0 − p 0  + µ 
+
0
 ∂ x ∂y
0 − p 

 ∂w ∂u
+

 ∂x ∂z
∂u ∂v
+
∂y ∂x
∂v
2
∂y
∂w ∂v
+
∂y ∂z
∂u
+
∂z
∂v
+
∂z
∂w 
∂x 

∂w 
.
∂y 
∂w 
2

∂z 
(23b)
The first matrix in (23) is the normal pressure stress. The second matrix
is the viscous stress tensor, proportional to the viscosity coefficient μ. The
diagonal elements of the viscous stress are associated with elongations of
fluid elements, and the off-diagonal elements are due to shearing deformations of the form shown in figure 3.2.
Most common fluids, including water and air, are found to conform to
the Newtonian stress relations (23) for all practical purposes. An exception occurs if dilute solutions of polymers are formed which disrupt the
isotropic nature of the fluid. The resulting stress relations are said to be
“non-Newtonian.” In certain cases the frictional drag on a body can be
reduced by this means, as described in the surveys by Lumley (1969) and
Bark, Hinch, and Landahl (1975).
The Motion of a Viscous Fluid
65
3.7 The Navier-Stokes Equations
It is now a simple matter to obtain the Navier-Stokes equations, which
express conservation of momentum for a Newtonian fluid, by substituting
the stress-strain relations (23) in Euler’s equations (18). The required derivatives of the stress tensor are
∂τ ij
∂p
∂  ∂ui ∂u j 
=−
+µ
+
∂x j
∂xi
∂xj  ∂xj ∂xi 
=−
(24)
∂p
∂2 ui
,
+µ
∂xi
∂x j ∂x j
since, from the continuity equation (15a),
∂2 uj
∂ ∂u j
=
= 0.
∂xj ∂xi ∂xi ∂xj
Thus, we obtain the desired Navier-Stokes equations:
∂ui
1 ∂p
1
∂u
∂2 ui
+ uj i = −
+ν
+ Fi ,
∂t
ρ ∂xi
∂x j
∂x j ∂x j ρ
where ν = μ/ρ is the kinematic viscosity coefficient. Written in vector form,
∂V
1
1
+ ( V ⋅ ∇ )V = − ∇p + ν∇2 V + F ,
∂t
ρ
ρ
(25b)
or, finally, in Cartesian coordinates:
∂u
∂u
∂u
∂u
1 ∂p
1
+u
+v
+w
=−
+ ν∇2 u + Fx
∂t
∂x
∂y
∂z
ρ ∂x
ρ
∂v
∂v
∂v
∂v
1 ∂p
1
+u
+v
+w
=−
+ ν∇2 v + Fy
∂t
∂x
∂y
ρ ∂y
ρ
∂z
∂w
∂w
∂w
1
∂w
1 ∂p
+u
+v
+w
=−
+ ν∇2 w + Fz .
∂t
∂x
∂y
∂z
ρ
ρ ∂z
(25c)
This system of three partial differential equations, together with the continuity equation (15), governs the motion of a viscous fluid subject only
to the assumptions of constant density and a Newtonian stress-strain relationship. The latter is justified for all practical purposes in the case of most
fluids, including water and air.
The difficulty comes in attempting to solve the Navier-Stokes equations;
they form a coupled system of nonlinear partial differential equations
and have been solved analytically only for some very simple geometrical
66
Chapter 3
configurations, principally those in which the nonlinear convective acceleration terms (V ⋅ ∇) V can be assumed to vanish. We shall examine some
examples of such flows subsequently.
3.8 Boundary Conditions
Before attempting to solve the Navier-Stokes equations, it is necessary to
impose appropriate physical conditions on the boundaries of the fluid
domain. In fact, it is precisely these conditions that distinguish different
flow problems.
The simplest boundary condition is that which must be imposed on a
solid boundary, such as the boundary between the fluid and a rigid body.
In this case, the fluid velocity equals the velocity of the body, since in a
viscous fluid the existence of a shear stress requires both the normal and
tangential velocity components of the fluid and the boundary to be equal.
An alternative physical boundary is a free surface, such as will exist
between the surface of the ocean and the atmosphere or between the
fluid and vapor domains of a cavitating flow. In these instances the fluid
density is so large, by comparison with the neighboring gas, that tangential
stresses are negligible; the only stress at such a surface is the normal pressure, which is generally known and specified (atmospheric pressure or vapor
pressure for the two cases cited). This is the dynamic boundary condition
at a free surface. In addition, it is necessary to impose the kinematic boundary condition that the normal velocity of the fluid and of the free surface
are equal.
3.9 Body Forces and Gravity
In our derivation of the Euler and Navier-Stokes equations, an external
body force per unit volume, F, has been included without specification or
restriction. The most common and important body force in the context
of marine hydrodynamics is the force due to gravity ρg, where g is the
gravitational acceleration vector. For all practical purposes g is a constant,
acting downward with magnitude 9.8 m/s2 or 32.2 ft/s2. If this is the only
body force acting upon the fluid, or more generally if the body force is
conservative, it can be represented as the gradient of a scalar function, in
the form F = ρ∇Ω ( x ) where, for gravity, Ω = g∙x. From the Navier-Stokes
The Motion of a Viscous Fluid
67
equations (25), this force can be absorbed with the pressure p, by means of
the substitution p = p − ρΩ . With this modification the Navier-Stokes equations no longer include the body-force terms explicitly. Hence, as far as
conservation of momentum is concerned, the only effect of a gravitational
force or any conservative force field is to change the pressure by an additive
amount −ρΩ.
Generally we regard p as the total pressure, p − ρΩ as the hydrodynamic
pressure, and the difference as the hydrostatic pressure. If the fluid is in a
state of rest, the total and hydrostatic pressures are equal.
In view of this additive decomposition, the body force has no effect on
the fluid motion, unless it affects the problem via a boundary condition.
Except for this possibility, the effect of gravity can be ignored throughout, with the understanding that this gives rise to an additive hydrostatic
pressure.
Gravity will play an important role when we consider wave motions
on the free surface. Here the dynamic boundary condition states that the
total pressure is equal to a specified value, and the hydrostatic component
must be included. When the boundary conditions are entirely kinematic,
however, with prescribed values of the normal velocity, there can be no
interaction between gravitational forces acting on the fluid and the dynamics of the fluid motion. With this justification, we will hereafter ignore body
forces in general and the gravitational force in particular until chapter 6,
where we consider the problem of waves on a free surface.
3.10 The Flow between Two Parallel Walls (Plane Couette Flow)
The simplest solutions of the Navier-Stokes equations are those pertaining
to the steady flow of a viscous fluid between two parallel walls separated by
a distance h, under the assumption that the horizontal dimensions of the
walls are very large compared to h. It is reasonable then to assume that the
flow is independent of the coordinates parallel to the walls and depends
only on the transverse coordinate. Let us suppose that the two walls occupy
the planes y = 0 and y = h, and that the motion is in the x-direction, driven
either by a pressure gradient or by the imposed uniform motion of one wall
relative to the other or by a combination of these two effects.
Let us suppose that the boundary y = 0 is at rest, while the boundary
y = h moves in the positive x-direction with velocity U. (Such a situation
68
Chapter 3
can be realized in general by means of a coordinate transformation that
brings the lower boundary to rest.) Since the motion is independent of
both the x- and z-coordinates, it follows from the continuity equation that
∂v/∂y = 0; thus, since v must equal zero at the two boundaries, this velocity
must be identically zero throughout the flow region 0 < y < h. The velocity
component w is considered to vanish since there is no driving mechanism
for a flow in the z-direction.
It follows that the fluid velocity vector is parallel to the x-direction, that
is, u is the only nonzero component of the velocity vector and the flow is
everywhere parallel to the x-axis. In addition, since the flow is independent
of the x-coordinate, the convective acceleration terms of the Navier-Stokes
equations are all equal to zero, and these equations reduce to the relatively
simple system
1 ∂p
∂2u
=ν 2 ,
ρ ∂x
∂y
∂p
= 0,
∂y
∂p
= 0,
∂z
(26)
subject to the boundary conditions u(0) = 0 and u(h) = U. From (26) the
pressure is independent of y and z, and ∂p/∂x = dp/dx is constant since u is
independent of x. The solution, obtained by integration, is
u=
1 dp
U
y( y − h ) + y.
2 µ dx
h
(27)
The pressure gradient gives rise to a parabolic profile, and the relative velocity between the two walls generates a linear shear flow. In general, a combination of these two flows will exist, as shown in figure 3.3.
3.11 The Flow through a Pipe (Poiseuille Flow)
A similar solution can be obtained for the flow in a tube of circular crosssection; for laminar flows, this solution is physically important for the
determination of pressure drops in pipes. Strictly speaking, the solution
should be obtained in circular cylindrical coordinates. To avoid the details
of transforming the Navier-Stokes equations into such a coordinate system,
we shall find the solution from the Cartesian form of the Navier-Stokes
equations, with the x-axis coincident with the axis of the tube; we assume
that u = u(r) depends only on the radial coordinate r, and hence on the
combination y2 + z2, while v = w = 0. The reduction of the Navier-Stokes and
The Motion of a Viscous Fluid
69
Figure 3.3
Couette flow between two parallel walls. The curves are labeled with values of the
nondimensional pressure gradient P = −(h2/2μU)(dp/dx). Note the occurrence of a
secondary backflow, analogous to separation, when this parameter is less than −1.
(From Schlichting 1968)
continuity equations is then identical to that carried out for plane Couette
flow. Hence the axial velocity satisfies the equation
1 dp
 ∂2 u ∂2 u 
= ν 2 + 2 ,
 ∂y
ρ dx
∂z 
(28)
subject to the boundary condition u(r0) = 0.
The appropriate solution of (28) is given by
u( r ) = −
1 dp 2
1 dp 2
( r0 − y 2 − z 2 ) = −
( r0 − r 2 ).
4µ dx
4µ dx
(29)
Once again, the pressure gradient dp/dx drives a flow parabolic in profile
across the tube. It is simple to compute the volume rate-of-flow or flux Q of
fluid through the pipe, by integrating across the circular cross-section from
y = 0 to y = r0,
r0
Q = 2π ∫ u( r )r dr =
0
π r04  dp 
−
.
8µ  dx 
(30)
70
Chapter 3
Reynolds and others have determined experimentally that such a flow
will exist in practice only if the Reynolds number based upon the diameter of the tube is less than a critical value.2 Otherwise, the flow becomes
unsteady or turbulent and is characterized by rapid velocity fluctuations in
both space and time. The comparatively smooth flow existing at Reynolds
numbers less than the critical value is a laminar flow.
3.12 External Flow Past One Flat Plate
The solutions obtained above illustrate steady laminar internal flow where
the fluid is restricted in width. In our subsequent consideration of external
flow, for example past a streamlined body in two or three dimensions, the
situation is significantly different. Not only may the form of the body be
relatively complicated, but also the fluid domain is unbounded in the direction normal to the body surface.
A technique for dealing with this type of problem was provided by
Prandtl’s introduction of the boundary-layer approximation. For sufficiently large values of the Reynolds number, it is reasonable to suppose
that within the bulk of the fluid, viscous forces will be negligible by comparison with inertial forces, and the corresponding flow may be considered
inviscid. This is a significant simplification, for as we shall see in the following chapters, various techniques can be employed to analyze inviscid
flows. Nevertheless, a consequence of viscous shear stresses is that however
large the Reynolds number may be, the fluid velocity on any rigid boundary of the fluid must still be equal to the velocity of the boundary. Thus,
there must exist significant viscous shear in a thin boundary layer at the
surface of any body that moves relative to the bulk of the fluid. To simplify
the Navier-Stokes equations, however, we can exploit the thinness of this
region.
This approach provides a scheme for calculating viscous effects, at least
for unseparated flows at high Reynolds number; equally important, it gives
a rationale for neglecting viscous stresses outside the boundary layer. Moreover, if the body is sufficiently regular in shape, its radii of curvature will be
much larger than the boundary-layer thickness, and the local flow within
the boundary layer will be effectively plane. Thus, if one imagines looking
at the boundary-layer flow, say by enlarging it with a magnifying glass or
The Motion of a Viscous Fluid
71
microscope, the details of the flow within this thin region will become visible, but the overall shape of the body is lost to view and the boundary of
the body will appear practically flat within the region of view.
This approach leads us to expect that the most important and relevant
viscous flow problem is the external flow past a single flat plate. This might
be taken to coincide with the plane y = 0, as for the lower wall of the plane
Couette flow, but with the fluid occupying the entire upper half-space
y > 0. We might hope to obtain the desired result from the solution (27)
of the corresponding problem for two plates, situated at y = 0 and y = h,
simply by letting the separation distance h tend to infinity while focusing
attention on the flow near the lower plate at y = 0. A cursory examination
of equation (27) shows that this will not work, however. The solution for
Couette flow is linear if driven by the movement of one plate and parabolic
if driven by a pressure gradient. In either event the presence of the second
boundary affects the flow near the first, and viscous effects persist throughout the entire fluid.
This analogy suggests that for steady laminar flow past a single infinitely
long flat plate, viscous effects will persist infinitely far away from the plate
in the normal direction. In principle this is correct and should not be surprising. Viscous effects are diffusive, and in this problem, diffusion has an
infinite amount of time to take effect. This is not very useful in our quest
for a boundary-layer theory where viscous effects are confined to a thin
region, and in some sense we must remove the opportunity for diffusion to
occur without limit in time and space.
Two possible modifications can be imposed to make this problem more
realistic physically and to limit the time for viscous diffusion to occur. First,
we might consider an unsteady problem where the time is limited by the
duration of the motion. Alternatiyely, for steady-state flows, the length of
the plate should be finite to limit the elapsed time during which a single
particle of fluid is traveling past the plate. The latter problem is the most
important and relevant one for us to consider, but it is also the more mathematically complicated. Thus, we shall first solve the unsteady problem for
a plate of infinite length, which is not without some practical interest in
any event.3
72
Chapter 3
3.13 Unsteady Motion of a Flat Plate
As we have assumed above, the plate coincides with the plane y = 0 and
the fluid occupies the half-space y > 0. If the plate moves in its own plane
parallel to the x-axis with prescribed velocity U(t), then the flow is again
independent of the x- and z-coordinates and the z-component of the
fluid velocity is zero. From continuity and the boundary condition on
the plate, it follows that ν = 0 and the velocity field is a parallel flow in
the x-direction which depends only on the coordinate y and on time t.
Thus from the Navier-Stokes equations, in the absence of any external
pressure gradients, the unknown velocity u(y, t) is governed by the partial
differential equation
∂u
∂2u
=ν 2.
∂t
∂y
(31)
This equation is called the diffusion equation, or heat equation, since it
governs the diffusion of heat in a conducting medium.
Appropriate boundary conditions are the no-slip condition on the plate,
u( 0, t ) = U (t )
on y = 0
(32)
and, in addition, the condition that the fluid motion tends to zero far away
from the plate,
u( y , t ) → 0
as y → ∞.
(33)
The last condition is somewhat pragmatic; it may be replaced by the initial
condition that the fluid is at rest everywhere, at some initial time before the
plate is set in motion. But (33) enables us to consider as the simplest possible example sinusoidal time dependence, which in fact violates the initial
condition. We shall return to this point subsequently.
For the simplest unsteady motion, assume that the motion of the plate
is sinusoidal, say U(t) = U 0 cos ωt where ω is the radian frequency. The
resulting flow is thus sinusoidal, and it is convenient to write the unknown
velocity u(y, t) in the complex form
u( y , t ) = Re[ f ( y )e iωt ],
(34)
where the real part of f is the velocity component in phase with cos ωt,
and the imaginary part is in phase with sin ωt. Substituting (34) in the
The Motion of a Viscous Fluid
73
governing equation (31) gives an ordinary differential equation for f in
the form
iω f = ν
d2f
.
dy 2
(35)
The solution is well known and can be written as
f = Ae ky + Be − ky ,
(36)
where the constants A and B are to be determined subsequently and
k = (iω / ν )1/2 = (1 + i )(ω / 2ν )1/2 .
(37)
(In view of the form of (36), one can take the positive or negative square
root, as this would only reverse the roles of the two constants A and B.)
Since the parameter k is complex with positive real part, the boundary
condition (33) at infinity can only be satisfied if A = 0 in (36). The boundary
condition (32) on the plate determines the value of the remaining constant
B, and the final solution of the problem is
u( y , t ) = Re{U 0 exp( − ky + iωt )}
= U 0 exp[ −(ω / 2ν )1/2 y ]cos[(ω / 2ν )1/2 y − ωt ].
(38)
The motion corresponding to (38) is a damped transverse wave; it oscillates in the x-direction while propagating in the y-direction away from the
plate. The wavelength is λ = 2π(2ν/ω)1/2, and this will be extremely short
for all but very low frequencies. For example, if ν = 10−6m2/s (for water), and
ω = 2π (1-second period), the wavelength is 0.35 cm. Moreover, the exponential factor in (38) will attenuate the wave amplitude in this distance
by a factor e−2π = 0.002. These numbers suggest that if the motion is not
extremely slow, the boundary layer will be very thin.
Since the governing equation (31) and boundary conditions (32–33)
are linear, more general time-dependent solutions can be constructed by
superposition. Moreover, if the frequency ω is permitted to be complex,
the motion of the plate will increase or decrease exponentially with time,
according as Im(ω) is negative or positive, respectively. The only restriction
here is that for the condition at y = ∞ to be satisfied, Re(k) must be positive
as assumed above. Hence if ω is complex, it must be negative imaginary corresponding in (34) to an exponentially growing velocity of the plate. Indeed,
this is the only reasonable alternative from the standpoint that the motion
74
Chapter 3
is initially at rest at t = −∞. Laplace transforms can then be used to solve
more general initial-value problems, including the particular case of a stepfunction plate velocity, which is initially at rest but impulsively accelerated
to a constant velocity U.
There is a simpler way to solve the latter problem, because no characteristic time scale is associated with the acceleration of the plate. Thus from
dimensional analysis, the unknown velocity u(y, t) must depend on only
one nondimensional similarity parameter, which will be taken as
η=
y
,
2(νt )1/2
(39)
with the time t measured from the instant when the plate is accelerated.
Using the chain rule to evaluate the derivatives in (31) gives the ordinary
differential equation
d2f
df
+ 2η
=0
dη 2
dη
(40)
for the nondimensional velocity f = u/U. The boundary conditions for the
function f(η) are that f(0) = 1 and f(∞) = 0. The solution of the differential
equation (40) subject to these boundary conditions is
η
f =
u
2
= 1 − erf(η ) = 1 −
e − η 2 dη.
U
√ π ∫0
(41)
The function erf(η) is known as the error function; its values can be determined from various tables (Abramowitz and Stegun 1964). From such a
source it is easy to plot the nondimensional velocity profile shown in
figure 3.4.
Let us now try to measure the boundary-layer thickness—that is, the width
of the region in which the fluid velocity is affected by the velocity of the
plate. Clearly this region extends to infinity, but the attenuation of the
velocity with increasing distance from the plate is quite rapid; for example,
the nondimensional velocity at a value of y/2(νt)1/2 = η = 2.0 is approximately
u/U = 0.005, a very difficult velocity to measure experimentally. A common
engineering criterion for the boundary-layer thickness is the distance from
the plate where the velocity is reduced to 0.01 of its value on the boundary.
This thickness is found from the above tables to be approximately
δ = 3.64(νt )1/2 .
(42)
The Motion of a Viscous Fluid
75
Figure 3.4
Velocity distribution near an impulsively accelerated wall. (From Schlichting 1968)
Again, the boundary-layer thickness is relatively small except over extremely
large time scales where, ultimately, the effects of diffusion are more widespread. For example, the boundary-layer thickness predicted from (42) for
water is 0.4 cm after one second and 0.24 m after one hour!
3.14 Laminar Boundary Layers: Steady Flow Past a Flat Plate
Let us turn now to the complementary problem of steady-state flow past a
flat plate of finite length l. Once again the motion is assumed two dimensional, independent of the z-coordinate, and the plate is taken to coincide
with the portion 0 < x < l of the plane y = 0. With w = 0, the continuity
equation and (steady-state) Navier-Stokes equations take the form
∂u ∂v
+
= 0,
∂x ∂y
u
∂u
∂u
1 ∂p
 ∂2u ∂2 u 
+v
=−
+ν 2 + 2 ,
 ∂x
ρ ∂x
∂x
∂y
∂y 
(43)
(44)
76
u
Chapter 3
∂v
∂v
1 ∂p
 ∂2v ∂2v 
+v
=−
+ v 2 + 2.
 ∂x
ρ ∂y
∂x
∂y
∂y 
(45)
The boundary conditions require that u = v = 0 on the portion of the
plane y = 0 occupied by the plate, say 0 < x < l, and that the velocity at
large distances from the plate must be equal to the undisturbed flow u = U,
v = 0.
The above equations are exact, regardless of the magnitudes of the
dimensional parameters U, l, and ν or the Reynolds number Ul/ν. Our interest is focused now on the approximate solution of these equations when
the Reynolds number is large, anticipating that the boundary layer will
be very thin under this circumstance. We begin by making a qualitative
estimate of the boundary-layer thickness, based on the unsteady solution
of the preceding section. If the characteristic time during which viscous
effects act on a fluid particle is t = l/U, the corresponding boundary-layer
thickness obtained from (40) will be δ = 3.64 (νl/U)1/2, or in nondimensional
form, δ/l = 3.64 R−l/2. A more careful analysis will reveal that the form of this
equation is correct, but the constant is slightly less than 5.0. Nevertheless,
the above estimate is sufficient for our present purposes because it suggests
the order of magnitude of the boundary-layer thickness.
Since u ≃ U immediately outside the boundary layer, it follows that
the horizontal velocity component u must increase from 0 to U over the
very small transverse distance δ. Therefore, the derivative ∂u/∂y will be
large, of order (U/δ), compared with changes parallel to the plate, of order
(U/l). Thus ∂u/∂y ≫ ∂u/∂x. From the continuity equation it follows that
∂u/∂y ≫ ∂v/∂y, or that the change in normal velocity v is relatively small in
passing across the boundary layer. Since v = 0 on the plate it follows that
this component of the velocity must be small throughout the boundary
layer, or u ≫ v.
As a result of these inequalities, the Navier-Stokes equations can be
simplified to the forms
u
∂u
∂u
1 ∂p
∂2u
+v
=−
+ν 2 ,
ρ ∂x
∂x
∂y
∂y
0=
1 ∂p
.
ρ ∂y
(46)
(47)
The Motion of a Viscous Fluid
77
Here the concept that u changes most significantly in the transverse direction has been used to discard the second derivative with respect to the
horizontal coordinate, or ∂2u/∂x2 ≪ ∂2u/∂y2.
From (47) it follows that the pressure p is not changed significantly in
passing through the boundary layer, as one might anticipate on physical
grounds. As a result, the longitudinal pressure gradient ∂p/∂x can be set
equal to its value outside the boundary layer. For a flat plate the inviscid
flow outside the boundary layer is uniform, with velocity U = constant.
Thus the pressure gradient ∂p/∂x = 0, and the laminar boundary-layer equations for a flat plate are the two relatively simple equations
u
∂u
∂u
∂2u
+v
− ν 2 = 0,
∂x
∂y
∂y
∂u ∂v
+
= 0.
∂x ∂y
(48)
(49)
It remains to nondimensionalize these equations. Taking advantage of
the disparate length scales, we stretch the transverse coordinate in proportion to δ; hence, appropriate independent variables are
x′ = x / l ,
y ′ = yR1/2 / l.
Then, guided by the continuity equation, we nondimensionalize the velocity components in the form
u′ = u / U ,
v ′ = vR1/2 / U .
With these substitutions, equations (48) and (49) are replaced by
u′
∂u ′
∂u ′ ∂ 2 u ′
+ v′
−
= 0,
∂x ′
∂y ′ ∂y ′ 2
∂u ′ ∂v ′
+
= 0,
∂x ′ ∂y ′
(50)
(51)
with the boundary conditions
u′ = v ′ = 0
u′ → 1
on y ′ = 0,
for y ′ → ∞.
Since (50–51) do not involve the viscosity coefficient or Reynolds number
explicitly, the solution of these equations will be a similarity solution valid
78
Chapter 3
for all Reynolds numbers, in the regime where the laminar boundary-layer
equations (48–49) are valid.
To reduce these partial differential equations to ordinary differential
equations, let us assume that the length l is large compared to the distance
downstream x and that the flow is unaffected locally by the actual length l.4
The solution of these equations then must be independent of l, so that the
dependence on the space coordinates x' and y' can only involve the ratio
y'/√x' = y (U/νx)1/2. Similarly, to remove the dependence on l in the defini-
tion of v', it follows that v' √x' must depend only on the ratio y'/√x'. Thus,
we seek a solution in the form
u
= F(η ),
U
v( x / U ν )1/2 = G(η ),
where η = y(U/νx)1/2. With these substitutions the resulting mathematical
problem is
1
dF
dF d 2 F
− ηF
+G
−
= 0,
2
dη
dη dη 2
(52)
1 dF dG
− η
+
= 0,
2 dη dη
(53)
F ( 0 ) = 0,
F( ∞ ) = 1,
G( 0 ) = 0.
These are coupled, ordinary differential equations for the two unknowns
F and G. The continuity equation (53) can be used to eliminate one
unknown, and in this manner we seek a stream function f(η) such that
F = df / dη,
G=
1  df

−f.
η

2  dη
Then the continuity equation is satisfied automatically, and the resulting
problem is as follows:
f
f =
df
=0
dη
d2f
d 3f
+ 2 3 = 0,
2
dη
dη
df
= 1 at
on η = 0,
dη
(54)
η = ∞.
Further progress requires that this nonlinear differential equation be
integrated numerically. The resulting solution is identified with Blasius,
who computed the boundary-layer solution for a flat plate in 1908. The
The Motion of a Viscous Fluid
79
resulting plot of the velocity distribution in the boundary layer is shown
in figure 3.5.
This figure is qualitatively similar to figure 3.4, for the velocity
distribution on an accelerating plate. Blasius’ solution predicts the value
u/U = 0.99 when η = 4.9, and the boundary-layer thickness as defined before
is approximately δ = 4.9(νx/U)1/2. An alternative definition of the boundarylayer thickness, which can be defined unambiguously, is the displacement
thickness
∞
δ * = ∫ (1 − u / U )dy.
(55)
0
Calculations reveal that
δ * = 1.72(ν x / U )1/2 .
(56)
This value is shown as a dotted line in figure 3.5 and corresponds graphically to the top edge of the rectangle, which has the same enclosed area as
the total area under the curve.
Figure 3.5
Velocity distribution in a Blasius laminar boundary layer. The total area under the
curve is equal to the area under the dashed line, corresponding to the flux defect and
the displacement thickness δ*.
80
Chapter 3
Physically, the displacement thickness is a useful concept since it defines
a region of width equal to the retardation of fluid flux in the boundary
layer, divided by the stream velocity U. Thus δ* represents the effective
amount by which the body is “thickened” because of the boundary layer,
and in this sense adding δ* to the body thickness will correct the body
shape to account for viscous effects. Corresponding to this loss in flux will
also be a loss in momentum, which defines the momentum thickness
∞
u
(1 − u / U )dy.
U
0
θ=∫
(57)
Numerical evaluation results in a value for the flat plate of
θ = 0.664(ν x / U )1/2 .
(58)
This description of the velocity distribution and boundary-layer thicknesses is independent of the x-coordinate and Reynolds number, by virtue
of the nondimensionalization of the y-coordinate as shown in figure 3.5.
Figure 3.6 shows these results in a dimensional form but with a greatly
magnified lateral scale. Here the development of the boundary layer with
distance downstream is more apparent. Upstream of the plate the velocity
profile is uniform, whereas downstream of the leading edge the velocity
profile of the boundary layer grows in the lateral direction proportional to
x1/2. The outer limit of the boundary layer is defined roughly by the 0.99
thickness parameter, whereas the displacement thickness represents the
outward displacement of the inviscid streamlines. In this figure, the lateral
scale is distorted by an amount that depends on the Reynolds number. To
Figure 3.6
Development of laminar boundary layer along a flat plate. The lateral scale is magnified as explained in the text.
The Motion of a Viscous Fluid
81
get some idea of the magnification, note that if the longitudinal length
scale is as represented in this figure full-scale, and if water were flowing
past with a velocity of 1 m/s, or 2 knots, the magnification factor for the
lateral dimension would be about 1:40. In other words, the displacement
thickness at the right-hand side of this figure would be about one quarter
of a millimeter.
Finally, we can consider the drag force exerted on the plate by viscous
shear stresses. The appropriate shear stress component is
τ xy = µ( ∂u / ∂y )y = 0 = µU (U / ν x)1/2 ( d 2 f / dη2 )η = 0 ,
(59)
or, after a calculation,
τ xy = 0.332 ρU 2 Rx −1/2 ,
(60)
where Rx = Ux/ν is the local Reynolds number based on distance from the
leading edge. The total drag, acting on one side of the plate, is
l
dx
= .664b( µρlU 3 )1/2 ,
1/2
x
0
D = .332 µbU (U / ν )1/2 ∫
(61)
where b is the width of the plate. The resulting frictional drag coefficient is
CF =
D
= 1.328(ν / Ul )1/2 = 1.328 R−1/2 .
1
2
ρU ( bl )
2
(62)
This curve is shown in figure 2.3 along with experimental values that support the theory in the laminar regime. Experimental confirmation for the
velocity distribution (figure 3.5) can be found in Schlichting (1968), for
Reynolds numbers ranging from 105 to 7 × 105. For Reynolds numbers
greater than 2 × 105 but less than about 3 × 106, the flow may be either
laminar or turbulent, depending on the degree of roughness of the plate
and also on the ambient turbulence level in the incident flow. When the
flow becomes turbulent, the Blasius solution is no longer valid and must
be replaced by one based upon the analysis of turbulent boundary layers.
3.15 Laminar Boundary Layers: Steady Two-Dimensional Flow
The analysis of the laminar boundary layer on a flat plate can be extended
to more general steady flows past two-dimensional bodies, such as the flow
82
Chapter 3
past a streamlined strut of large aspect-ratio. If the radius of curvature of
the body surface is large compared to the boundary-layer thickness, the
same analysis carries over independent of the gradual change in direction
due to the curvature of the body surface. The boundary-layer equation (46),
together with the continuity equation (43), remains valid in this case provided that x and y are taken to be locally tangential and normal coordinates at each point of the body. However, the pressure gradient ∂p/∂x is
now significant, since the inviscid flow will be affected by the body and
will possess a nontrivial inviscid pressure distribution. Thus, one must first
determine the inviscid flow past the body, compute the corresponding pressure gradient on the body surface, and then attempt to solve the boundarylayer equation (46) with the pressure gradient included. If more accuracy is
desired, this procedure can be carried further by recalculating the inviscid
flow, including the additional effects of the displacement thickness due to
the first-order boundary layer, and then finding a second-order boundarylayer solution.
In view of this synthesis, we can discuss the qualitative features of the
two-dimensional boundary-layer theory for general two-dimensional flow
by discussing the flow past a flat plate with an arbitrary external pressure
gradient imposed. For the usual situation of the flow past a streamlined
body, the relevant pressure typically will be a maximum at the leading and
trailing edges, where there are stagnation points, and will fall to a minimum value along the sides of the body where the velocity is a maximum.
Thus, a typical pressure gradient will include a negative region over the
forward portion of the body and a positive region over the afterbody.
The most significant feature of the pressure gradient is its effect upon the
boundary layer and, especially, the circumstances under which it can lead
to separation. If separation occurs at some point on the body surface, the
streamlines will break away from the body at this point, enclosing a separated wake downstream of relatively high vorticity and low pressure. The
separation point, if it exists, will be a stagnation point at which the flow
from ahead and behind meets and abruptly changes direction. Ahead of
this point the tangential velocity will be positive, and behind it the velocity will be negative, as shown in figure 3.7. It is clear from this figure that
separation will occur when the shear stress is zero, or
( ∂ u / ∂ y ) y = 0 = 0.
(63)
The Motion of a Viscous Fluid
83
Figure 3.7
Flow past a body with separation. The second figure shows the local flow near the
separation point, in terms of local coordinates that are tangential (x) and normal (y)
to the body at the separation point. The graphs show the different velocity profiles,
in the boundary layer, for a favorable pressure gradient on the forebody (A), an unfavorable pressure gradient on the afterbody (B), at the separation point (C), and
downstream of the separation point (D). Note the qualitative similarity for each of
these velocity profiles with the Couette flows shown in figure 3.3.
It is also apparent that a necessary condition is that
( ∂ 2 u / ∂ y 2 ) y = 0 > 0.
This requirement can be related to the pressure gradient from the boundarylayer equation (46). For this purpose we set y = 0; thus u = v = 0, and we
obtain from (46) the relation
µ
∂ 2 u ∂p
=
∂y 2 ∂x
on
y = 0.
(64)
This qualitative discussion is obviously not sufficient to determine the
drag due to shear stresses acting on the body surface. Before describing an
approximate computational method, we first note that from conservation
84
Chapter 3
of momentum in the boundary layer, the shear stress on the body surface
can be related to the displacement thickness, momentum thickness, and
the pressure gradient of the velocity field outside the boundary layer, in a
simple manner due to von Kármán. For this purpose we integrate equation
(46) in the normal direction, across the boundary layer:
δ
 ∂u
∂u 
δ
δ
0
0
1 ∂p
∂2u
∫  u ∂x + v ∂y  dy = − ρ ∫ ∂x dy + ν ∫ ∂y
0
2
dy.
(65)
Here the upper limit of integration δ is somewhat arbitrary; ultimately we
shall require only that it be sufficiently large that the velocity attains its
asymptotic value u = U. From Bernoulli’s equation (4.21) for the pressure
outside the boundary layer,
p=−
1
ρU 2 + constant.
2
(66)
Thus the pressure gradient (constant across the boundary layer) is
∂p
dU
.
= − ρU
∂x
dx
(67)
Now, regrouping the terms in (65),
δ
 ∂u
∂u
∫  u ∂x + v ∂y − U
0
δ
∂2u
dU 
dy = ν ∫ 2 dy

∂y
dx 
0
 ∂u 
 ∂u 
= ν  −ν 
 ∂y  y = 0
 ∂ y  y =δ
(68)
= −τ xy / ρ ,
since ∂u/∂y → 0 outside the boundary layer. Next we integrate the continuity equation in a similar manner to obtain
y
 ∂u
∂v 
∫  ∂x + ∂y  dy = 0
(69)
0
or, since v = 0 on the boundary y = 0,
y
∂u
dy.
∂x
0
v ( x, y ) = − ∫
(70)
Substituting this expression for v in (68), and integrating by parts, we obtain
The Motion of a Viscous Fluid
85
y
−τ xy δ  ∂u ∂u  ∂u 
dU 
= ∫ u
−
dy  − U
 dy
∫

ρ
∂x ∂y  0 ∂x 
dx 
0
δ
δ
dU 
∂u
∂u
∂u
−U
= ∫ u
+u
 dy − U ∫ ∂x dy
∂
x
dx
x
∂

0
0
δ
∂u
∂u
dU 
= ∫ 2u
−U
−U
dy
∂
x
∂
x
dx 

0
δ
(71)
δ
∂
dU
(U − u )dy
[ u(U − u )]dy − ∫
∂
x
dx
0
0
= −∫
=−
δ
δ
d
dU
(U − u )dy.
u(U − u )dy −
dx ∫0
dx ∫0
Recalling that the upper limit of integration δ is a point outside of the
boundary layer, and using the definitions of the displacement and momentum thickness from (55) and (57), we obtain the following result:
τ xy
d
dU
=
(U 2θ ) + U δ *
.
ρ
dx
dx
(72)
For the flat plate with zero pressure gradient, the last term of this equation is zero, and (72) can be integrated with respect to x to give a relation
between the drag coefficient and the momentum thickness. This relation is
consistent with equations (58) and (61) and expresses the physical relationship that exists between the hydrodynamic force acting on the plate and
the loss of momentum in the fluid.
Finally, we shall describe a scheme, due to Pohlhausen, for computing
the characteristics of a two-dimensional laminar boundary layer with a
pressure gradient. We suppose that the velocity distribution in the boundary layer can be approximated by a fourth-degree polynomial, in the form
( u / U ) = a( y / δ ) + ( y / δ )2 + c( y / δ )3 + d( y / δ )4 .
(73)
Here the condition that u = 0 on the boundary y = 0 has been satisfied
by eliminating the constant term in the polynomial. The remaining four
boundary conditions that can be imposed on physical grounds are that on
the inner boundary y = 0, the boundary layer equation (46) holds, or
ν
∂ 2 u 1 ∂p
dU
, on y = 0,
=
= −U
2
ρ ∂x
∂y
dx
and that, on the outer boundary,
86
Chapter 3
u = U,
∂u
= 0,
∂y
∂2 u
= 0, on y = δ .
∂y 2
(The first derivative ∂u/∂y must vanish if u is to approach U asymptotically,
and the second derivative then vanishes as a consequence of equation
(46).) These four conditions determine the four constants a, b, c, and d in
equation (73) in terms of the thickness parameter δ. If we introduce the
nondimensional parameter
Λ=
δ 2 dU
,
ν dx
(74)
the four coefficients in (73) are determined as
a = 2 + Λ / 6,
b = − Λ / 2,
c = −2 + Λ / 2,
d = 1 − Λ / 6.
The velocity distribution within the boundary layer is approximated
then by a one-parameter family of curves that depends only on the shape
factor Λ. A plot of this family appears in figure 3.8. Certain values of the
shape factor Λ have special significance. The value Λ = 0 corresponds to a
boundary layer with zero pressure gradient, i.e., the case of the flat plate
treated in the preceding section. Positive values of Λ correspond to favorable (decreasing) pressure gradients as described in figure 3.8, and negative
values of Λ correspond to unfavorable pressure gradients. The criterion (63)
for inception of separation corresponds to the value Λ = −12; larger negative
values suggest the reverse flow region further downstream. Positive values
of Λ greater than 12 are excluded, since these imply a velocity within the
boundary layer larger than that outside.
For different values of the shape factor Λ, values of the displacement and
momentum thickness and of the shear stress can be computed with the
following results:
δ
u
Λ 
3
δ * = ∫  1 −  dy = δ 
,
−

 10 120 
U
(75)
0
δ
u
u
Λ
Λ2 
 37
,
1 −  dy = δ 
−
−
 315 945 9, 072 
U
U
0
θ=∫
(76)
µU 
Λ
 ∂u 
τ xy = µ  
=
2+ .
 ∂y  y =0
6
δ 
(77)
The Motion of a Viscous Fluid
87
Figure 3.8
Pohlhausen velocity profiles, for various values of the shape factor Λ.
If these values are substituted in the momentum equation (72), we obtain
a differential equation involving both Λ(x) and δ(x), but the latter can be
eliminated by using the definition (74). The final result of these substitutions is the following differential equation for Λ:
d Λ dU / dx
d 2U / dx2
=
g( Λ) +
h( Λ ),
U
dU / dx
dx
(78)
where
g( Λ ) =
7257.6 − 1336.32 Λ + 37.92 Λ 2 + 0.8Λ 3
,
213.12 − 5.76 Λ − Λ 2
h( Λ ) =
213.12 Λ − 1.92 Λ 2 − 0.2 Λ 3
.
213.12 − 5.76 Λ − Λ 2
Further progress requires that this differential equation be integrated
numerically with respect to x, for any given inviscid velocity function
U(x), starting upstream at the stagnation point where the boundary layer
commences on the body surface. At the stagnation point, we must have
U = 0, while dU/dx > 0; to avoid a singularity at this point from the first
88
Chapter 3
term on the right side of equation (78), g(Λ) must be zero. Of the three possible roots, the appropriate one is Λ = 7.052, and this criterion determines
the initial value of the shape factor Λ for numerical integration of the differential equation (78). The corresponding initial velocity distribution is
indicated in figure 3.8.
As a simple estimate of the accuracy of this method, the results may
be compared with Blasius’ solution for the flat plate, corresponding to the
curve Λ = 0 in the above method. The displacement and momentum thicknesses from Pohlhausen’s approximation are found to be
δ * = 1.75(ν x / U )1/2 ,
(79)
θ = 0.685(ν x / U )1/2 .
(80)
Thus, the error incurred in this case by assuming the polynomial profile
(73) is, by comparison with equations (56) and (58), approximately 2 to 3
percent.
3.16 Laminar Boundary Layers: Closing Remarks
Three-dimensional laminar boundary layers are complicated not only by
the changes in the external pressure gradient but also by cross-flow components in the third dimension within the boundary layer, and simple
qualitative conclusions cannot be made. An exceptional case is the axisymmetric flow past a body of revolution, where a transformation due to
Mangler reduces the continuity equation to its two-dimensional form and
preserves the form of the boundary-layer equation. Unsteadiness is another
complication, but it is of less practical concern for most problems of marine
hydrodynamics. Discussions of these and other aspects of laminar boundary layers can be found in the references by Landau and Lifshitz (1959),
Batchelor (1967), Schlichting (1968), Yih (1969), and White (1974).
3.17 Turbulent Flow: General Aspects
Turbulence is characterized by the superposition of a highly irregular and
oscillatory velocity pattern upon an otherwise “smooth” flow. Thus, flows
that would otherwise be steady in time and slowly varying in space become
unsteady and rapidly fluctuating. If the Navier-Stokes equations are to
The Motion of a Viscous Fluid
89
remain valid under these conditions, the temporal and spatial variations
of the velocity vector must be included. On the other hand, the principal characteristics of practical importance are quantities such as the mean
velocity profile or the mean drag acting on a body, and this suggests that
one should average the velocity, stresses, etc., with respect to space or time.
With this in mind, we can separate the velocity vector into mean and fluctuating parts as follows:
ui = ui + ui′.
(81)
Here, a bar denotes the mean or average value, and ui′ the fluctuating
component.
This average may be defined with respect to time, space, or an ensemble
of occurrences. If the intent of separating turbulent and mean flow components is to be realized, the average with respect to time must be taken
over a temporal period that is large compared to the characteristic period
of turbulent oscillations but small compared to the time scale of the basic
flow. Similarly, spatial averages must be carried out over length scales small
compared to the length scale of the basic or mean flow. If the basic flow is
steady-state, a time-average is the simplest concept to adopt.
If we substitute the velocity components into the Navier-Stokes and continuity equations and then average these equations, we can progress toward
a decomposition of the overall problem. Before doing so, we recall some
familiar averaging properties of oscillatory systems. First, the average of the
velocity ui is, by definition, ūi. Thus, from (81),
u′i = 0,
(82)
with a similar result for derivatives of the velocity,
∂ui ∂ui
=
;
∂x j ∂x j
∂ui′
= 0.
∂x j
(83)
On the other hand, for a product of two velocities, or derivatives of the
velocities, it is necessary to include the average of the products of the fluctuating components, such as
ui u j = ( ui + ui′)( u j + u′j ) = ui u j + ui′u′j .
(84)
Now let us examine the consequences of substituting (81) in the continuity equation (15a),
90
Chapter 3
∂ui ∂ui′
+
= 0.
∂xi ∂xi
(85)
If averages are taken, this equation applies independently to the mean
velocity,
∂ui
= 0,
∂xi
(86)
and hence, by subtraction of (86) from (85), also to the fluctuating
components:
∂ui′
= 0.
∂xi
(87)
If we follow the same procedure with the Navier-Stokes equations (25),
there is a contribution to the averaged equations from the nonlinear terms,
involving the fluctuating components. Thus
∂ui
1 ∂p
∂u
∂u ′
+ u j i + u′j i = −
+ ν∇2 ui .
∂t
ρ ∂xi
∂x j
∂x j
(88)
Using the continuity equation, we can transform the additional contribution represented by the third term in (88) as follows:
u′j
∂u′j
∂ui′
∂
ui′u′j − ui′
=
∂x′j ∂xj
∂x j
=
∂
ui′u′j .
∂x j
(89)
If this is transferred to the right side of (88), we obtain
∂ui
∂u
1 ∂p
∂
+ uj i = −
+ ν∇2 ui −
ui u j .
∂t
∂x j
ρ ∂xi
∂x j
(90)
If (90) is regarded as an equation for the mean velocity ūi, the only effect
of the velocity fluctuations is to contribute an additional stress system.
By comparison with Euler’s equations (18), this Reynolds stress is equal to
− ρui′u′j . In other words, the total stress tensor can be written in the form
 ∂u ∂u j 
τ ij = − pδ ij + µ  i +
− ρui′u′.
j
 ∂xj ∂xi 
(91)
The Motion of a Viscous Fluid
91
Physically, the Reynolds stress can be identified with the transport of
momentum associated with the fluctuating components of the velocity. If
this stress were known, equation (91) could be solved for the mean velocity ūi in a manner analogous to the solution of the Navier-Stokes equations for laminar flow. However, there is no rational method of determining
the Reynolds stress system, for example by imposing additional boundary
conditions, and so it is necessary to resort to empirical methods based on
experimental observations.
In general, Reynolds stresses are more important than the ordinary
viscous stresses. This is apparent from (91) since the Reynolds number is
inversely proportional to the viscosity coefficient μ. Thus, the Reynolds
stress system will be dominant at high Reynolds numbers, except in the
viscous sublayer very close to the boundaries, where the mean velocity gradient is large and, simultaneously, the fluctuating components are forced to
vanish by the boundary condition ui′ = 0 on the wall.
The increased stress level due to the Reynolds stress suggests that the
frictional drag on a body will be greater in the turbulent regime than is
the case for laminar flow, and this is confirmed by experimental data, as
shown in figure 2.3. An alternative physical explanation is suggested by
the convection of momentum, represented by the Reynolds stress. Since
this momentum convection will cause the velocity profile to become more
uniform, as shown in figure 3.9, it must result ultimately in a larger velocity
gradient and shear stress on the body surface.
3.18 Turbulent Boundary Layer on a Flat Plate
Now let us consider the characteristics of the mean velocity distribution for
the flow in a turbulent boundary layer. To keep this very complex problem
as simple as possible, we shall assume a flat plate with zero pressure gradient. Also, we shall be concerned only with the mean velocity ū, and the bar
will be deleted with the understanding that the average value is implied.
The tangential velocity component u is assumed to be of the form
u = f ( x, y , ρ , µ ,U ),
(92)
where x and y are the usual space coordinates with the plate occupying the
positive x-axis, ρ and μ are the density and viscosity of the fluid, and U is
the velocity of the free stream outside the boundary layer. The subsequent
92
Chapter 3
Figure 3.9
Comparison of laminar (———) and mean turbulent (-----) velocity profiles, for
the boundary layer on a flat plate. Since the boundary layer thickness is substantially greater in the turbulent case, the difference in scales should be noted in this
comparison.
development of nondimensional variables is facilitated if the coordinate x
is replaced by the boundary-layer thickness δ, under the assumption that
dδ/dx > 0, so that to each value of x corresponds a unique value of δ and vice
versa. The exact definition of the boundary-layer thickness is not important; it will be regarded somewhat loosely as the value of y where u = U,
even though this equality is only asymptotic. Similarly, the velocity U will
be replaced by the friction velocity,
uτ ≡ [τ xy ( x, 0 ) / ρ ]1/2 ≡ (τ 0 / ρ )1/2 .
(93)
Again, to justify this· substitution of variables, we rely on a physical argument: for each value of U and fixed x, there is a unique value of the shear
stress on the wall, and vice versa.5 With these substitutions, the functional
form of equation (92) is replaced by
u = f (δ , y , ρ , µ , uτ ).
(94)
After nondimensionalization, it follows that
u
= f ( uτ y / ν , y / δ ).
uτ
(95)
The Motion of a Viscous Fluid
93
Since Reynolds stresses dominate viscous stresses except in the viscous
sublayer, viscous stresses identified with the viscosity coefficient ν and
hence the parameter uτy/ν in (95), are significant only in the immediate
vicinity of the boundary y = 0, where the turbulent fluctuations and associated Reynolds stress approach zero. In this viscous sublayer, y/δ ≪ 1 and
(96) may be replaced by the inner law or law of the wall
u
uy
uy
f  τ , 0 ≡ f 1  τ  .
 ν

 ν 
uτ
(96)
In other words, the nondimensional velocity very close to the wall depends
only on the Reynolds number based on the friction velocity uτ and the
distance from the boundary y. This conclusion is not surprising since, in
the viscous sublayer, the turbulent outer portion of the boundary layer is
remote, and the boundary-layer thickness δ is relatively unimportant compared to the viscosity coefficient, shear stress, and distance from the wall.
On the wall y = 0, two boundary conditions are appropriate. First we
impose the usual condition u = 0; hence f1(0) = 0 in (96). In addition, it follows from the definition of the shear stress that
 ∂u 
 df ( u y / ν ) 
µ   = τ 0 = µuτ  1 τ
 = τ 0 f1′ ( 0 ).
∂
y
dy
  y =0

 y =0
(97)
When the shear stress is related to the friction velocity by means of (93),
f1 must have a linear behavior as y → 0, of the form
f1( uτ y / ν ) uτ y / ν.
(98)
Now let us consider the outer region of the boundary layer, where the
Reynolds stress dominates the viscous stress. In this region we expect the
parameter y/δ to be important, in conjunction with the outer “boundary
condition” that u = U when y/δ = 1. This suggests the appropriate form for
the outer approximation of (95), or the velocity-defect law
U −u
= f 2 ( y / δ ).
uτ
(99)
The velocity difference has been used to impose the outer boundary condition in the form
f 2 (1) = 0.
(100)
94
Chapter 3
Moreover, equation (99) reflects the significant fact that in the outer region,
momentum flux is balanced by the Reynolds stress and the former depends
not on the absolute velocity u but on the velocity defect U − u.
To amplify briefly on the justification for the velocity-defect law, let us
view the flow in a frame of reference moving with the free stream. Here
the relative fluid velocity is ur = u − U, which can be expressed in the nondimensional form (99). In these terms, U now enters only in the boundary
condition on the plate y = 0, where ur = −U. However, in terms of the flow in
the outer region, the plate is deeply embedded within the viscous sublayer;
its effect on the momentum balance in the outer region is represented by
the shear stress at the wall, and hence by the friction velocity uτ. On this
basis the relative velocity ur in the outer region must depend only on uτ and
y/δ, as assumed in (99).
At this point two complementary expressions have been obtained for
the velocity in the turbulent boundary layer; the inner approximation (96)
is valid in the viscous sublayer, and the outer approximation (99) is valid in
the main portion of the boundary layer where the Reynolds stress is dominant. Neither of these is a complete solution, of course, but they at least
represent simplifications of the overall problem compared with the more
general velocity (95) which depends on two parameters. Moreover, further
progress can be made with the apparently bold assumption that there exists
an overlap region where both approximations are valid simultaneously—that
is, where
f1( uτ y / ν ) = U / uτ − f 2 ( y / δ ).
(101)
This seems at first to be an unjustifiable assumption, since the inner
expression f1 has been derived by neglecting the Reynolds stress, whereas
the outer function on the right side of (101) is assumed to hold if and
only if the Reynolds stress is dominant. The principal justification for
(101) is that it leads to conclusions that compare well with experiments,
namely a logarithmic velocity profile that will be derived later. First, however, a mathematical justification for (101) can be found from the concept
of matched asymptotic expansions. Thus, the inner approximation should
hold when y/δ ≪ 1, and the outer approximation if uτy/ν ≫ 1. Can both
these inequalities hold simultaneously for some suitable values of the distance from the wall y? In other words, can an intermediate value of y be
found such that ν/uτ ≪ y ≪ δ? Clearly this is possible, provided ν/uτ ≪ δ or
The Motion of a Viscous Fluid
95
uτδ/ν ≫ 1. The latter is a Reynolds number, which represents the ratio of
inertial forces to viscous forces in the outer portion of the boundary layer.
By assumption this Reynolds number is large, and hence an overlap region
exists where (101) is valid.
It is now a simple matter to show that the functions f1 and f2 must be
logarithmic in the overlap region. Differentiating both sides of (101) with
respect to y and multiplying the result by y, we obtain
( uτ y / ν )f1′( uτ y / ν ) = −( y / δ )f 2′( y / δ ),
(102)
where a prime denotes the derivative with respect to the appropriate argument of each function. The left side of (102) is a function only of the Reynolds number uτy/ν, while the right side depends only on the ratio y/δ. Since
these two variables are independent, the functions on both sides of the
equation can be equal to each other if and only if they are equal to a constant, say A. Thus,
f1′( uτ y / ν ) = A(ν / uτ y )
and
f 2′( y / δ ) = − A(δ / y )
or, after integrating,
f1 = A log( uτ y / ν ) + C1 ,
(103)
f 2 = − A log( y / δ ) + C2 ,
(104)
where C1 and C2 are constants of integration.
These equations are valid only in the overlap region. Indeed, it would be
remarkable if (103) and (104) represented the complete solution throughout the boundary layer, for they have been derived from the boundary conditions alone; the Navier-Stokes equations and Reynolds stress have been
used only for qualitative arguments. Thus the inner velocity function fl
cannot satisfy the inner boundary condition, u = 0 on y = 0, with any choice
of the constants A and C1, and the outer function f2 cannot satisfy the outer
condition u = U when y = δ, unless C2 = 0. The latter condition was invoked
originally by Prandtl, but this does not permit a satisfactory empirical fit
within the overlap region.
Thus we have arrived at the conclusion, originally due to Prandtl and
von Kármán, that the mean velocity profile is logarithmic in the intermediate or overlap region. Much effort has gone into corroborating these
formulas and determining the constants A, C1, and C2. Surveys of these
investigations and more complete expositions of the above derivation are
96
Chapter 3
given by Schlichting (1968) and by Monin and Yaglom (1971). Figures 3.10
and 3.11, reproduced from the latter reference, show comparisons with
experiments for the inner and outer functions respectively; the constants
there have been chosen empirically to give the following equations in place
of (103) and (104):
u
= 2.5 log( uτ y / ν ) + 5.1,
uτ
U −u
= −2.5 log( y / δ ) + 2.35,
uτ
uτ y / ν > 30,
y / δ < 0.15.
(105)
(106)
Figure 3.10
Universal dimensionless mean velocity profile of turbulent flow close to a smooth
wall according to the data of tube-, channel- and boundary-layer measurements.
Here u+ = u/uτ and z+ = uτy/ν. (From Monin and Yaglom 1971)
The Motion of a Viscous Fluid
97
Figure 3.11
Logarithmic form of the velocity defect law for a boundary layer. Here u* = uτ and
z = y. (From Monin and Yaglom 1971)
Natural logarithms are implied here and subsequently, except where explicitly stated in (115). Figures 3.10 and 3.11 both display the restriction that
the logarithmic profiles are correct only in the intermediate region, and
the respective domains of validity are given in (105–106). Figure 3.10 also
shows the region of validity of the near-wall approximation (98), which is
valid for uτy/ν < 5. The experimental data in figure 3.10 includes results not
only for a flat plate boundary layer, but for various tubes and channels as
well, since the inner velocity profile is the same for all these cases.
An equation for the friction velocity uτ, and hence the shear stress on the
wall, may be deduced by adding (105) and (106) to give the relation
U
uδ
= A log τ + (C1 + C2 ).
uτ
ν
(107)
However, a second equation is needed to relate the two unknowns uτ and
δ. For this purpose, we recall the momentum equation (72), which was
derived for a laminar boundary layer. It is possible to show that this same
equation is valid for a turbulent boundary layer by including the Reynolds
stress in the boundary-layer equations, provided the fluctuating components of the velocity satisfy the isotropic relation u′ 2 = v ′ 2 . With this additional assumption, and deleting the pressure gradient term in (72), we get
98
Chapter 3
τ 0 = ρU 2 dθ / dx = ρuτ 2 ,
(108)
where (93) has been used, and θ denotes the momentum thickness. The
boundary-layer thicknesses can be calculated in terms of the two definite
integrals
1
1
 U − u
I = ∫ f 2 ( y / δ )d( y / δ ) ∫ 
d( y / δ ),
 uτ 
0
0
1
1
(109)
2
 U − u
J = ∫ [ f 2 ( y / δ )]2 d( y / δ ) ∫ 
d( y / δ ).
 uτ 
0
0
(110)
Except for the comparatively thin viscous sublayer, these approximations
are valid, and the integrals in (109–110) must have universal constant values independent of the Reynolds number. Thus, from the definitions (55)
and (57),
δ * / δ = I ( uτ / U ),
(111)
θ / δ = I ( uτ / U ) − J ( uτ / U )2 .
(112)
Equations (107), (108), and (112) give a total of three equations for the
three unknowns uτ, δ, and θ; these can be solved, without further assumptions, in terms of the constants that can be determined empirically. One of
the principal results of this procedure is the local frictional-drag coefficient
1
cf = τ 0
ρU 2 , which satisfies the implicit equation
2
1
= 2 −1/2 A log( Rx c f ) + C3 ,
√ cf
(113)
where Rx = Ux/ν is the local Reynolds number. The total drag coefficient
l
CF =
1
c f ( x )dx
l ∫0
(114)
can be determined from a similar equation, and this is the basis for Schoenherr’s flat-plate frictional-drag formula
1
= 1.79 log( Rl CF ) = 4.13 log10 ( Rl CF ).
√ CF
(115)
This important result is displayed in figure 2.3, which also shows that
experimental confirmation exists over a broad range of Reynolds numbers.
The Motion of a Viscous Fluid
99
The lower limit for this range of validity is where transition from laminar
flow occurs, generally between Rl = 2 × 105 and 2 × 106. The upper limit is
undetermined; the maximum Reynolds number where experimental data
exist is about 4.5 × 108. In this range of two to three decades, the Schoenherr
formula (115) is apparently a satisfactory fit to the experimental data. Nevertheless, some uncertainty remains from the standpoint of ship-resistance
predictions, insofar as large, high-speed ships operate at Reynolds numbers between 109 and 1010. The Schoenherr formula also has been criticized
at the lower limit of Reynolds numbers, where towing-tank tests are carried out on small ship models. This subject is discussed by Matheson and
Joubert (1974) and by Granville (1975).
3.19 The 1/7-Power Approximation
A simpler velocity distribution, but one that has less empirical support
and scientific motivation than the logarithmic profile, is the 1/7-power
relation
u / uτ = 8.7( uτ y / ν )1/7 .
(116)
Despite its shortcomings, this formula is useful as a basis for qualitative
conclusions. For example, if we invoke the condition that u = U when y = δ
and solve for the shear stress on the boundary, it follows that
τ 0 = ρuτ 2 = 0.0227ρU 2 (U δ / ν )−1/4 .
(117)
A direct calculation of the momentum thickness gives
δ
δ
u
(1 − u / U )dy = ∫ ( y / δ )1/7[1 − ( y / δ )1/7 ]dy
U
0
0
θ=∫
(118)
= (7 / 8 − 7 / 9)δ = 0.0972δ .
Then from (108) and (117), it follows that δ must satisfy the differential
equation
0.0972
dδ
= 0.0227(U δ / ν )−1/4 .
dx
(119)
Multiplying both sides by δ1/4, and integrating, we have
δ 5/4 = 0.292(ν / U )1/4 x + C ,
(120)
100
Chapter 3
where the constant of integration must be determined from the initial
thickness δ at the point of transition from laminar to turbulent flow. If the
Reynolds number is sufficiently high, or if turbulence is induced near the
leading edge by studs or other devices, then we can set C = 0 in (120) and
obtain the relations
δ = 0.373 x Rx −1/5 ,
(121)
θ = 0.0363 x Rx −1/5 ,
(122)
τ 0 = 0.0290 ρU 2 Rx −1/5 ,
(123)
δ * = 0.0467 x Rx −1/5 .
(124)
Here Rx = Ux/ν, and the last result for the displacement thickness δ* follows
from a straightforward integration from 0 to δ using (55).
By comparison with the laminar boundary layer, which increases in
thickness with distance downstream at a rate proportional to x1/2, the turbulent boundary layer has a faster rate of growth, proportional to x4/5 according to the 1/7-th power assumption.
3.20 Roughness Effects on Turbulent Boundary Layers
So far we have assumed that the flat plate is hydraulically smooth. Marine
vessels may be roughened either by features of their fabrication, such as
welding seams or rivet heads, or by the inevitable marine fouling that accumulates in service. Thus, it is appropriate to discuss the effects of roughness
and the extent to which a distinction can be made between rough and
smooth surfaces.
To introduce roughness into the boundary-layer theory, we must define
a scale of roughness, say the height k of the roughness elements. The significance of the roughness can be estimated then in terms of the magnitude
of k relative to the dimensions of the boundary layer. If k is small compared
to the height of the viscous sublayer, the plate is effectively smooth and the
frictional drag independent of k. In the other extreme, if k is large compared
to δ, the boundary layer will be unimportant by comparison to the form
drag resulting from the roughness elements, which in turn will be essentially independent of the Reynolds number.
The Motion of a Viscous Fluid
101
For a fixed scale of roughness, in terms of the characteristic length l of
the plate, k/l = constant. In this case the frictional drag is identical to that of
a smooth plate below a Reynolds number which depends on the roughness
ratio; above this point, the frictional drag is effectively constant, as shown
in figure 3.12. From this result we can infer that smoothness of a full-scale
ship hull is much more important than smoothness of its model, for if the
two had the same relative roughness or equal values of k/l, the effects of the
roughness would increase with Reynolds number.
On the other hand, for constant values of the Reynolds number based
on the roughness scale, Uk/v, corresponding to k/δ = constant, there is a
family of curves approximately parallel to the smooth curve. The latter are
more representative of the ship roughness, insofar as the roughness scale of
ship hulls of different lengths is effectively independent of the length. In
this sense a roughness allowance can be approximated by a constant additive
resistance coefficient such as 0.0004, as discussed in chapter 2.
Figure 3.12
Resistance formula of sand-roughened plate; coefficient of total skin friction. (From
Schlichting 1968)
102
Chapter 3
3.21 Turbulent Boundary Layers: Closing Remarks
Semiempirical methods have been developed for analyzing the effects of a
pressure gradient. A critique of this subject is given by Thompson (1967).
Qualitatively, the effects of the pressure gradient are analogous to the case
of a laminar boundary layer. In particular, negative and positive values of
the pressure gradient in a streamwise direction along the body surface are
respectively favorable and unfavorable for the stability of a thin boundary
layer, as compared with the occurrence of separation. Turbulence serves the
useful role of convecting more momentum from the free stream into the
boundary layer, by comparison with the laminar case, and thus the separation point is delayed. In this connection, we recall the drag coefficient of
a sphere, shown in figure 2.2, which is decreased by the effects of turbulence at the transition Reynolds number of approximately 400,000. Similar
remarks apply to the development of separation or stall on lifting surfaces;
even in the normal regime of small angles of attack, the pressure gradient
can significantly affect the turbulent boundary layer.
Semiempirical extensions of the turbulent boundary-layer theory have
been developed to account for the presence of external pressure gradients
and of three-dimensional effects. Extensive discussion of these topics is
given in the monograph by Cebeci and Smith (1974). Applications to the
calculation of the viscous drag on a ship hull are discussed by Huang and
von Kerczek (1972) and Kux (1974).
Problems
1. The Couette flow shown in figure 3.3 is a rough model for the flow in a
lubricated bearing. Find the bearing friction per unit area, in terms of the
viscosity of the lubricant and the gap width of the bearing. How does the
bearing friction change if there is a pressure gradient in the direction of
motion?
2. The simplest equation of motion for the spar buoy shown in figure 2.16
is mÿ + ρgS0(y − A cos ωt) = 0 where y(t) is the heave amplitude, S0 is the
waterplane area, and A is the wave amplitude. Use the solution in section
3.13 for sinusoidal motion of a flat plate to derive a linear damping coefficient proportional to the submerged area of the cylindrical portion of the
spar buoy. Using this estimate of the damping coefficient, find the resonant
The Motion of a Viscous Fluid
103
response for a spar buoy of diameter 1m and draft 5 m in waves of amplitude 0.5 m.
3. Modify the analysis of section 3.10 for the case where the upper boundary is a free surface with no stress. Include the component of the hydrostatic
pressure acting in the direction of the flow and find the surface velocity of a
fluid of depth 1cm flowing down an inclined plane of angle 1 degree. Based
on this steady-state solution, what is the surface current in the Cape Cod
Canal, where the depth is 10 m, the length is 10 km, and the maximum
difference in elevation between the two ends is 3 m? Give several reasons
why this estimate is exaggerated.
4. Reformulate the problem of the Cape Cod Canal as stated in problem 3,
assuming that this motion is sinusoidal in time with period 12 hours and
that the velocity is horizontal and a function only of the vertical coordinate. Show that the solution is of the form
u( y , t ) = Re[ e iωt ( ae ky + be − ky + c )].
Determine the constants in this equation, the maximum velocity on the
surface, and the phase shift between the current and the tidal difference at
the two ends. (The actual current in this canal is about five knots.)
5. For a Reynolds number of 106, compute the displacement thickness
and shear stress at the boundary for laminar (Blasius) and turbulent (1/7
power) boundary layers. Use these results to sketch the two velocity profiles, as was done in figure 3.9, but with the vertical scale (y) preserved in
both curves.
6. For a 2 m ship model moving at 1.0 m/s, estimate the displacement thickness of the boundary layer at the stern, using the Blasius result for Laminar
flow, and the 1/7 power law for turbulent flow. Compare the resulting values of δ*/l with the corresponding turbulent result for a 200 m ship moving
at 10 m/s.
7. Given the differences between salt and fresh water at the same temperature, how would the boundary-layer thickness and shear stress of a ship
change as it moves from salt water into fresh water, if the ship’s velocity
and the water temperature are unchanged?
8. A common speed-measuring device for small boats is a paddlewheel that
protrudes about 1cm from the hull surface and rotates about an axis flush
104
Chapter 3
with the hull surface in response to the fluid moving past the hull. If such
a paddlewheel is mounted 3 m downstream from the bow, estimate the
linearity of this device over speeds ranging between 1 and 10 m/s.
9. Comment on the validity of the following statement: Barnacles and similar marine growth are unimportant on the surface of a ship, since the size
of these organisms is negligible relative to the size of the ship. Using figures
2.11 and 3.12, estimate the effects of barnacles, of typical scale k = 1 mm,
on the full-scale resistance of the Lucy Ashton at a speed of 15 knots. How
much will this speed be reduced assuming equal total drag in the smooth
and rough conditions?
References
Abramowitz, M., and I. A. Stegun, eds. 1964. Handbook of mathematical functions.
Washington: U.S. Government Printing Office.
Bark, F. H., E. J. Hinch, and M. T. Landahl. 1975. Drag reduction in turbulent
flow due to additives: A report on Euromech 52. Journal of Fluid Mechanics
68:129–138.
Batchelor, G. K. 1967. An introduction to fluid dynamics. Cambridge: Cambridge
University Press.
Cebeci, T., and A. M. O. Smith. 1974. Analysis of turbulent boundary layers. New York:
Academic.
Granville, P. S. 1975. Drag and turbulent boundary layer of flat plates at low Reynolds
numbers. David W. Taylor Naval Ship Research and Development Center Report
4682.
Huang, T. T., and C. H. von Kerczek. 1972. Shear stress and pressure distribution on
a surface ship model: Theory and experiment. In Ninth symposium on naval hydrodynamics, eds. R. Brard and A. Castera, pp. 1963–2010. Washington: U.S. Government
Printing Office.
Kux, J. 1974. Three-dimensional turbulent boundary layers. In Tenth symposium on
naval hydrodynamics, eds. R. D. Cooper and S. W. Doroff, pp. 685–703. Washington:
U.S. Government Printing Office.
Landau, L. D., and E. M. Lifshitz. 1959. Fluid mechanics. English translation. London:
Pergamon.
Lumley, J. L. 1969. Drag reduction by additives. Annual Review of Fluid Mechanics
1:367–384.
The Motion of a Viscous Fluid
105
Matheson, N., and P. N. Joubert. 1974. A note on the resistance of bodies of revolution and ship forms. J. Ship Res. 18:153–168.
Monin, A. S., and A. M. Yaglom. 1971. Statistical fluid mechanics. English translation.
Cambridge, Mass.: The MIT Press.
Schlichting, H. 1968. Boundary layer theory. English translation. 6th ed. New York:
McGraw-Hill.
Thompson, B. G. J. 1967. A critical review of existing methods of calculating the turbulent
boundary layer. Ministry of Aviation, Aeronautical Research Council Reports and
Memoranda 3447.
White, F. M. 1974. Viscous fluid flow. New York: McGraw-Hill.
Yih, C. S. 1969. Fluid mechanics. New York: McGraw-HilI.
4 The Motion of an Ideal Fluid
Chapter
The
Motion
of
an
Ideal
4
Fluid
© Massachusetts Institute of TechnologyAll Rights Reserved
Having obtained an understanding of viscous effects, we can justify their
neglect in certain circumstances. Thus, when the boundary layer is thin,
the bulk of the fluid is effectively inviscid or ideal and can be analyzed from
the Navier-Stokes equations with the viscous stress tensor deleted. At first
glance the resulting simplification of these equations is less apparent than
might be expected from physical considerations. Ultimately, the mathematical simplifications are considerable, and a wide variety of inviscid flow
problems are amenable to solution.
The fundamental equations are the continuity equation and Euler’s
equations, derived in chapter 3. Thus, the velocity vector V = (u1, u2, u3)
must satisfy the equations
∂ui / ∂xi = 0,
(1)
and
∂ui
∂u
1 ∂p 1
+ Fi ,
+ uj i = −
∂t
∂x j
ρ ∂xi ρ
i = 1, 2, 3.
(2)
In the second equation only the normal pressure stress is included, on the
assumption that the fluid is inviscid.
Hereafter, the external force field Fi will be assumed to consist only of
the gravitational force ρg which acts vertically downward. If x2 is the vertical coordinate and is positive in the upward direction, then Fi = (0, −ρg, 0).
Under these circumstances equation (2) can be written as
∂ui
∂u
1 ∂
( p + ρ gx2 ).
+ uj i = −
∂t
∂x j
ρ ∂xi
(3)
108
Chapter 4
In the simplest case of hydrostatic equilibrium, the velocity is zero (or a
constant) and p + ρg x2 is a constant.
4.1 Irrotational Flows
To simplify these equations, the class of possible fluid motions must be
restricted. For this purpose we define the circulation Γ as the integrated tangential velocity around any closed contour C in the fluid,
Γ = ∫ ui dxi .
(4)
C
Kelvin’s theorem of the conservation of circulation states that for an ideal
fluid acted upon by conservative forces (e.g., gravity) the circulation is
constant about any closed material contour moving with the fluid. Physically, this happens because no shear stresses act within the fluid; hence it
is impossible to change the rotation rate of the fluid particles. Thus, any
motion that started from a state of rest at some initial time, will remain
irrotational for all subsequent times, and the circulation about any material1
contour will vanish.
To prove Kelvin’s theorem, we consider the derivative of (4) with respect
to time, dΓ/dt. Since the contour C moves with the fluid particles, or with
velocity V, it follows that
 ∂
d
dΓ
∂ 
= ∫ ui dxi = ∫  + u j
( ui dxi ).
dt
dt C
t
xj 
∂
∂

C
(5)
Here the differential operator acting upon the integrand is the substantial
derivative D/Dt defined in equation (3.19), since the contour of integration
C is a material contour moving with the fluid particles. The resulting derivatives of the velocity components ui are straightforward to compute, but
some care is required for the differential element dxi. Since C is a material
contour, the differential element will itself be a function of time; to analyze
its resulting distortion, we resort to the definition of the integral as the limit
of a finite sum. Thus, if xi( n ) denotes the coordinates of the nth point along
the contour C, say with n = 1, 2, …, N, (5) can be replaced by
 ∂
dΓ
∂ 
= lim ∑
uiδ xi( n ) ,
+ uj
dt N →∞ n  ∂t
∂xj 
(6)
The Motion of an Ideal Fluid
109
provided that as N → ∞,
δ xi( n ) ≡ xi( n +1) − xi( n ) → 0.
Since the coordinates xi( n ) move with individual fluid particles, they
must be functions of time. By definition the velocity components at the
same points are given by
ui( n ) = ∂xi( n ) / ∂t .
(7)
Using the chain rule in (6), noting that the coordinates xi( n ) depend on time
but not on the space coordinates, and finally using (7), we obtain
 ∂ui

dΓ
∂u 
= lim ∑
+ u j i  ( xi( n +1) − xi( n ) ) + ui (ui( n +1) − ui( n ) ).
dt N →∞ n  ∂t
∂x j 

(8)
In this form, the limit is given by the integrals
 ∂ui
dΓ
∂u 
=
+ u j i  dxi + ∫ ui dui ,
dt C∫  ∂t
∂x j 
C
(9)
or, using the equations of motion (3),
dΓ
1 ∂
1
=− ∫
( p + ρ gx2 )dxi + ∫ d( ui ui ).
dt
ρ C ∂xi
2C
(10)
The right side of equation (10) is the integral of a perfect differential over
a closed contour and is, therefore, equal to zero. In effect, the integral can
be integrated to give
−
B
1
1
p + ρ gx2 − ρu j u j  ,

ρ
2
A
where B and A are the upper and lower limits of integration; since these
points are identical, the difference [
]BA = 0.
Since dΓ/dt = 0, the circulation must be a constant. We shall assume
hereafter that the fluid motion started from an initial state of rest, say when
t = −∞, so that Γ = 0 for all time and for all material contours within the
fluid.
Finally, we invoke Stokes’ theorem for a continuous differentiable
vector V,
∫∫ (∇ × V ) ⋅ dS = ∫ V ⋅ dx,
S
C
(11)
110
Chapter 4
where the surface integral is taken over any surface S bounded by the contour C. The line integral is by definition the circulation, which must equal
zero for any material contour lying within the fluid. Thus, the surface integral must vanish for any surface S, situated in the fluid, and this can be true
only if the integrand is identically zero. Hence the motion of the fluid is
irrotational,
∇ × V = 0,
(12)
or, in Cartesian components,
∂u ∂v
−
= 0,
∂y ∂x
∂v ∂w
−
= 0,
∂z ∂y
∂w ∂u
−
= 0.
∂x ∂z
(12a)
The left side of (12) is the vorticity, or the curl of the velocity, and (12a)
gives the Cartesian components of the vorticity. To say that the flow is
irrotational is equivalent to saying that the vorticity is zero throughout
the fluid. This conclusion is important because an irrotational vector can
be represented as the gradient of a scalar. This statement is the result of
Helmholtz' theorem in vector analysis, which states that any continuous
and finite vector field can be expressed as the sum of the gradient of a scalar
function ϕ plus the curl of a zero-divergence vector A. The vector A vanishes
identically if the original vector field is irrotational. The general proof of
this theorem can be found in Morse and Feshbach (1953). For our present
purposes, a simpler approach will suffice to show that if the velocity field is
irrotational, it can be represented simply as the gradient of the scalar function ϕ, or the velocity potential.
4.2 The Velocity Potential
Here we wish to show that if the fluid motion is irrotational, the velocity
vector V can be represented by the gradient of a scalar potential ϕ which
will depend generally on the three space coordinates xi and time t. Replacing the velocity by its potential may at first seem an unnecessary complication, since we can envisage the velocity and measure it with suitable
instruments in the laboratory, whereas the velocity potential is no more
than a mathematical abstraction. However, the velocity is a vector quantity
with three unknown scalar components, whereas the velocity potential is
a single scalar unknown from which all three velocity components may be
computed.
The Motion of an Ideal Fluid
111
Before proceeding to exploit the velocity potential, we must prove first
that it is indeed a valid representation of the unknown velocity vector. Our
proof will display clearly the importance of assuming that the motion is
irrotational. In particular, let us consider the definite integral
φ( x, t ) =
x
∫ u dx ,
i
i
(13)
x0
where the lower limit is some arbitrary constant position x0 and the upper
limit is the point x = (x1, x2, x3). This integral is independent of the particular path of integration between the points x0 and x, since the difference in
the value of any two integrals, between the same two points, is equal to the
circulation around the closed path from x0 to x along one path and back to
x0 along the other path, which is equal to zero if the fluid motion is irrotational. Thus, the integration in (13) can be performed along any desired
path. If we choose a path that approaches the point x along a straight line
parallel to the x1-axis, then along this final portion of the path of integration uidxi = u1dx1 so that
x
∂φ
∂
=
u1dx1 = u1.
∂x1 ∂x1 x∫0
(14)
The remaining portion of the integral, being a constant, does not contribute to the derivative. Applying a similar argument for the other two coordinates, we have in general that
ui = ∂φ / ∂xi ,
(15)
or
V = ∇φ .
(15a)
Thus, one can assume the existence of a velocity potential, subject only to
the requirement that the fluid motion is irrotational.
If (15) is substituted for the velocity vector in the continuity equation
(1), it follows that
∂ ∂φ
∂ 2φ
=
= 0,
∂xi ∂xi ∂xi ∂xi
or
(16)
112
Chapter 4
∇ 2φ ≡
∂ 2φ ∂ 2φ ∂ 2φ
+
+
= 0.
∂x 2 ∂y 2 ∂z 2
(16a)
This is the Laplace equation which expresses conservation of fluid mass for
potential flows and provides the governing partial differential equation to
be solved for the function ϕ.
It can be confirmed that the fluid motion defined by the velocity (15) is
irrotational, by recalling from vector analysis that the curl of a gradient is
identically zero or simply by substituting (15) in (12).
4.3 Bernoulli’s Equations
We have not yet exploited Euler’s equations or conservation of momentum, except to prove Kelvin’s theorem. Nor have we discussed the pressure,
which is related to the fluid velocity by the three Euler equations (2).
In two important cases the Euler equations can be integrated to give an
explicit equation for the pressure known as the Bernoulli integral, or simply
Bernoulli’s equation. The first result is perhaps the most familiar one from
elementary courses in fluid mechanics and pertains to steady flow without
assumption about the vorticity. The second case is irrotational flow, but it
includes the possibility of unsteadiness. The two proofs are different and
will be presented separately.
First we consider steady, but possibly rotational, flow. Here Euler’s equations take the form
uj
∂ui
∂
=−
( p / ρ + gx2 ).
∂x j
∂xi
(17)
Multiplying both sides by the velocity components ui and summing over
i gives
ui u j
∂ui
∂
= −ui
( p / ρ + gx2 ).
∂x j
∂xi
(18)
The left side of this equation can be rewritten as
1
∂
1
∂
uj
ui ui = ui
ujuj .
2 ∂x j
2 ∂xi
(19)
The Motion of an Ideal Fluid
113
In the last step, we have interchanged the two indices i and j; this has no
effect on the summation over both. Combining the last two equations and
collecting terms on one side gives the result
ui
∂ 
1
p / ρ + u j u j + gx2  = 0.

∂xi 
2
(20)
The differential operator ui∂/∂xi in (20) is the steady form of the substantial derivative, or the rate of change following a fluid particle with velocity
V. Thus along any streamline, whose tangent is everywhere parallel to V, the
quantity in parentheses in (20) is constant, and
p=−
1
ρV 2 − ρ gx2 + C ,
2
(21)
where
V 2 = ui ui = u2 + v 2 + w 2 .
(22)
In this result the constant C may vary for different streamlines, and a steady
motion independent of time has been assumed.
The alternative form of Bernoulli’s equation, valid for unsteady irrotational flows, is more useful since the flow of an inviscid fluid is generally
irrotational but may be unsteady. In this case we utilize the velocity potential, substituting (15) in the general form of Euler’s equations (3),
∂ ∂φ ∂φ ∂ ∂φ
1 ∂
( p + ρ gx2 ).
+
=−
ρ ∂xi
∂t ∂xi ∂xj ∂xj ∂xi
(23)
Direct differentiation shows that
∂φ ∂ ∂φ 1 ∂ ∂φ ∂φ
=
.
∂xj ∂xj ∂xi 2 ∂xi ∂xj ∂xj
(24)
Thus (23) can be rewritten in the form of a perfect differential
∂  ∂φ 1 ∂φ ∂φ 
1 ∂
=−
( p + ρ gx2 ).
+
∂xi  ∂t 2 ∂xj ∂xj 
ρ ∂xi
(25)
Integrating this with respect to the three space variables xi gives
∂φ 1 ∂φ ∂φ
1
+
= − ( p + ρ gx2 ) + C(t ).
∂t 2 ∂x j ∂x j
ρ
(26)
114
Chapter 4
The “constant” in (26) is independent of the space coordinates but may
depend on time. This constant can be absorbed into the velocity potential
by redefining the latter as
t
φ ′ = φ − ∫ C(t )dt .
(27)
Since the difference between ϕ and ϕ' is a function only of time, the gradient is not affected and either of these functions is an equally suitable velocity potential from the standpoint of (15). Substituting (27) in (26) leaves the
form of Bernoulli’s equation unchanged in terms of the velocity potential
ϕ', but the constant C(t) no longer appears. In other words, the value of C(t)
is related to a corresponding constant in the velocity potential which has
no effect on the velocity vector, and C(t) may be chosen arbitrarily. C(t) can
be set equal to zero and deleted entirely from Bernoulli’s equation, or it
can be set equal to some desired value such as the atmospheric pressure or
other relevant ambient pressure. In the remainder of this chapter, we shall
delete C(t) entirely.
Both forms of Bernoulli’s equation, (21) and (26), include in common
the dynamic pressure −
1
ρV 2 and the hydrostatic pressure −ρgx2. The
2
former expresses that the pressure is reduced in regions of high velocity
in accordance with the well-known Venturi effect. The hydrostatic term,
anticipated in section 3.9, represents the simple increase in pressure with
depth, equal to the static weight per unit area of the fluid above this point.
4.4 Boundary Conditions
The distinction between different types of fluid motions results from the
conditions imposed on the boundaries of the fluid domain. Thus we must
relate the physical conditions on these boundaries to appropriate mathematical statements.
As in chapter 3, two different types of boundary conditions must be discussed: a kinematic condition corresponding to a statement regarding the
velocity of the fluid on the boundary and a dynamic boundary condition
corresponding to a statement about the forces on the boundary. Unlike
the viscous case, there are now fewer conditions to impose, because in the
present case there are no shear stresses acting within the fluid.
The Motion of an Ideal Fluid
115
A kinematic boundary condition is appropriate on any boundary surface
with a specified geometry and position. In the simplest case the boundary
may be fixed and rigid, as at the sides and bottom of a stationary container
of fluid. In the more general instance, say of a rigid body moving with prescribed velocity U through the fluid, the velocity of this surface is nonzero.
In either case the physically relevant boundary condition for the fluid flow
at the boundary is that the normal component V·n of the fluid velocity
must be equal to the normal velocity U·n of the boundary surface itself. In
other words, no fluid can flow through the boundary surface. Expressed in
terms of the velocity potential, this condition becomes
∂φ / ∂n = U ⋅ n ,
(28)
where ∂/∂n denotes the derivative in the direction of the unit normal n
directed out of the fluid.
In the absence of viscous shear stresses, only the normal component of
the fluid velocity is prescribed on the boundary. A tangential velocity difference between the fluid and the boundary surface is not only permitted but
also expected. In a real fluid the conditions described are relevant immediately outside the boundary layer, as opposed to the actual body surface.
However, the distinction between these two positions is not significant to
the formulation of the inviscid flow problem, on the presumption that the
boundary-layer thickness is very small.
If the normal velocity (28) is prescribed on all the boundary surfaces
of a simply-connected fluid domain and the position of these surfaces is
known, then the solution of Laplace’s equation for the velocity potential is
specified uniquely except for an additive constant. The proof of this statement is sufficiently simple to repeat here. Let us suppose that two solutions
ϕ1 and ϕ2 both satisfy the same boundary condition (28) and Laplace’s equation throughout the fluid. The difference, ϕ3 = ϕ1 −ϕ2, is a homogeneous
solution with zero normal derivative on the boundary surface S and also a
solution of Laplace’s equation throughout the fluid volume  . By a wellknown consequence of the divergence theorem (3.5),
∂φ
∫∫ φ ∂n dS = ∫∫ (φ∇φ ) ⋅ dS = ∫∫∫ ∇ ⋅ (φ∇φ )d
S
S

= ∫∫∫ [ ∇φ ⋅ ∇φ + φ∇2φ ]d  .

(29)
116
Chapter 4
Under the specified conditions, substituting the potential ϕ3 gives
0 = ∫∫∫ (∇φ3 ⋅ ∇φ3 )d .
(30)

Since the integrand of this volume integral is positive definite, it must
vanish everywhere throughout the fluid. Hence there is no fluid motion if
there is no normal velocity on the boundaries, and the potential ϕ3 can be
at most a constant.
From this uniqueness proof it follows that the kinematic boundary condition (28) contains precisely the right amount of information regarding
the fluid motion. To prescribe more information on the boundaries would
be to overspecify the solution. On the other hand, there will be problems
where the position and velocity of the boundary are unknown, in particular on the free surface where waves occur and the form of the wave
motion, and free surface elevation, is not known a priori. In this instance
the kinematic boundary condition (28) remains valid, but the velocity U
of the boundary is unknown. Additional information must be provided,
and the physically relevant condition is that the pressure on the free surface is equal to atmospheric pressure. Thus the dynamic boundary condition is based on Bernoulli’s equation (26), with the pressure p prescribed.
Further discussion of this situation is deferred until we discuss free-surface
flows in chapter 6.
4.5 Simple Potential Flows
The simplest example of ideal fluid motion is the uniform stream, say with
velocity U flowing in the x-direction. The velocity potential of this flow is
clearly ϕ = Ux. More generally the components of similar streaming flows in
the y- and z-directions can be superposed to obtain the velocity potential
ϕ = Ux + Vy + Wz. This potential is a solution of Laplace’s equation (16), and
operating with the gradient (15) confirms that this is indeed the potential
of a flow with velocity components (U, V, W).
This streaming flow will be disturbed by the presence of a body or bodies or of some other type of boundary geometry such as the variable form
of a flow channel. From the mathematical viewpoint such a disturbance
requires the introduction of singularities; the simplest of these are the source
potential and its compatriot, the sink. Since these singularities are literally a
The Motion of an Ideal Fluid
117
source or sink of fluid, they violate the continuity condition and Laplace’s
equation locally at the point where they are located. This is permissible provided the singularity is situated within the body, or at most on the boundary surfaces, and is not allowed within the interior of the fluid.
The velocity potential of a source, situated at the origin, is
φ=
−m 2
m
( x + y 2 + z 2 )−1/2 = − ,
4π
4π r
(31)
where r is the radial distance from the origin, or from the position of the
source, and m is a constant. Since the source potential (31) depends only
on the coordinate r, its gradient will be a vector in the radial direction,
and hence the velocity field will consist of radial streamlines. It is straightforward to confirm that (31) is a solution of the Laplace equation (16) for
r ≠ 0; hence this radial flow satisfies the requirement of conservation of
fluid mass everywhere except at the singular point r = 0.
If the source potential (31) is differentiated with respect to r to obtain the
radial velocity and integrated over the surface of a sphere centered upon
the origin, it follows that the rate of flux Q of fluid emitted from the source
is precisely equal to m. This parameter is known as the source strength. If m
is negative, the flux direction is reversed, and the singularity is called a sink.
Mathematically, the distinction between a source and a sink is simply the
sign of the strength m; generally we shall use the term “source” without
distinction.
The streaming flow past a semi-infinite half-body can be developed by
superposing the source potential (31) and a free stream so that
φ = Ux −
m 2
( x + y 2 + z 2 )−1/2 .
4π
(32)
The resulting flow is axisymmetric about the x-axis, and the streamlines
in the x-y plane are as shown in figure 4.1. Differentiation of (32) with
respect to x indicates a stagnation point at x = −(m/4πU)1/2, where V = 0. Here
the flow is deflected around the source; thereafter the outer flow upstream
of the stagnation point continues downstream, but with a permanent
deflection from the x-axis due to the fluid emanating from the source.
Although the inner flow does not correspond to the physical domain of the
fluid, it is of interest because it reveals how the source serves to generate
the body. Thus, fluid originally emitted from the source tends to oppose
118
Chapter 4
Figure 4.1
Streaming flow past a semi-infinite half-body generated by a point source at the
origin. The body surface is axisymmetric about the x-axis and corresponds to the
position of the dividing streamlines.
the incoming stream and produces the stagnation point, but ultimately all
of the inner flow is diverted downstream to infinity. Since the rate of flux
emitted by the source is m, and since far downstream this fluid must move
with velocity U for the pressure to balance across the dividing streamline,
the cross-sectional area of the dividing streamline far downstream is equal
to m/U. The resulting half-body is semi-infinite in extent.
To represent the more practical situation where the body is closed and
of finite length, we need to introduce not only a source but also a sink of
opposite strength, located so that the fluid emitted by the source will be
absorbed into the sink. The sink clearly must be located downstream of the
source, and if these two singularities are situated symmetrically about the
origin, at x = ∓ a , the potential will be
φ = Ux −
m
m
[( x + a)2 + y 2 + z 2 ]−1/2 +
[( x − a)2 + y 2 + z 2 ]−1/2 .
4π
4π
(33)
Differentiation with respect to x, with y = z = 0, reveals that the stagnation
points are located at x = ±l/2, where the length l is determined from the
equation
(l 2 / 4 − a2 )2 = 2al( m / 4πU ).
(34)
The Motion of an Ideal Fluid
119
Figure 4.2
Streaming flow past a Rankine ovoid, or source-sink combination.
The streamlines associated with (33) define a Rankine ovoid, as shown in
figure 4.2. The maximum radius b can be found from continuity, since the
flux passing across the plane x = 0, inside a circle of radius b, will be equal
to the flux emitted from the source. Thus, with x = 0,
b
2am / 4π 

2π ∫ U + 2
R dR = m,
+ R2 )3/2 
(
a

0
(35)
or
m = πUb2 (1 + b2 / a2 )1/2 .
(36)
The resulting flow, shown in figure 4.2, is similar to that actually observed
for streamlined axisymmetric bodies. From Bernoulli’s equation one can
compute the pressure distribution on the body; it will take a maximum
value at the two stagnation points and a minimum at the central plane
x = 0 where the velocity is a maximum.
We might proceed to construct more general axisymmetric bodies by
distributing sources and sinks continuously along the body axis. This is
a practical method for determining the flow characteristics of bodies of
revolution, especially if they are relatively slender, as will be shown in chapter 7. Instead, let us focus our attention on the opposite extreme, where
the separation between the two singularities shrinks to zero. To preserve a
120
Chapter 4
finite effect from the two singularities as they are brought together, their
strengths m must increase at the same time, for otherwise they will cancel
out in the limit when they coincide. Thus it is necessary to make the product μ = 2ma a constant, with the result
µ
{[( x − a)2 + y 2 + z 2 ]−1/2 − [( x + a)2 + y 2 + z 2 ]−1/2 }
8π a
µ ∂
{[( x − a)2 + y 2 + z 2 ]−1/2 }a = 0
= Ux +
4π ∂a
µx
.
= Ux +
4π ( x2 + y 2 + z 2 )3/2
φ = Ux + lim
a→ 0
(37)
The last term is called a dipole or doublet, and the constant μ is its moment.
If we examine the resulting flow, from the combination of this dipole with
the uniform stream, in a spherical coordinate system where x = r cos θ and
(y2 + z2)1/2 = r sin θ, (37) takes the form
φ = Ur cos θ +
µ cos θ
.
4π r 2
(38)
Since the derivative with respect to the radius r vanishes on r = (μ/2πU)1/3,
(37) and (38) give the flow of a uniform stream, of velocity U, past a sphere
of this radius.
Two-dimensional singularities can be constructed analogously. Thus, if
the flow is independent of the z coordinate, or parallel to the x-y plane, the
source and dipole potentials are
m
log( x2 + y 2 )1/2 (source ),
2π
x
µ
( dipole).
2
2π x + y 2
Once again, a source will combine with a uniform stream to generate a
half-body, a source and equal sink will generate an elongated closed body,
and a point dipole will give the limiting and important case of the flow past
a circle of radius R = (μ/2πU)1/2 in the form
φ = Ux +
µx
.
2π ( x2 + y 2 )
(39)
The Motion of an Ideal Fluid
121
4.6 The Stream Function
In the preceding sections the concept of the velocity potential has been
used exclusively, as a functional representation of the flow field, and justified on the basis that the fluid velocity field is irrotational. Alternatively,
the velocity vector can be represented in terms of the stream function, whose
existence is instead a consequence of the continuity equation. For any
incompressible fluid, with or without viscosity, the velocity vector V must
satisfy the condition that its divergence is equal to zero; a well-known result
of vector analysis is that any divergence-free vector can be written in the
form
V = ∇ ×Ψ ,
(40)
where Ψ is the vector stream function. In general, this is not a very useful
concept, since the new unknown function is also a vector. However, in the
special cases of two-dimensional plane flow and three-dimensional axisymmetric flow, the vector stream function has only one component; thus it
becomes a scalar unknown, with the same resulting simplification as the
velocity potential.
First, let us consider two-dimensional plane motion in the x-y plane. The
velocity component w = 0 and the stream function should be independent
of z. These conditions can be met only if Ψ = (0,0,ψ) and the scalar stream
function ψ determines the velocity components u and v in accordance with
the relations
u = ∂ψ / ∂y ,
v = −∂ψ / ∂x.
(41)
The axisymmetric case follows from analogous reasoning. Thus, if we
introduce circular cylindrical coordinates (R,θ,x) and assume that the flow
is independent of θ, it follows that the stream function vector consists only
of a component in the θ-direction, which is independent of that coordinate. It is desirable in this case to define Ψ = (0, ψ/R, 0) so that the radial
and axial velocity components are given by
uR = −
1
∂ψ / ∂x ,
R
ux =
1
∂ψ / ∂R.
R
(42)
(The form of the curl operator, in circular cylindrical coordinates, is given
in Hildebrand, 1976, section 6.18.)
122
Chapter 4
The derivation of the stream function can be supplemented by a physical argument. For plane flow in the x-y plane, we define the scalar, which is
a measure of the flux across a contour C, as
ψ = ∫ ( udy − vdx).
(43)
C
The contour C is shown in figure 4.3. From continuity, (43) is independent
of the choice of C; as in the analogous integral (13) defining the velocity
potential, (43) depends only on the two endpoints, say x0 and x, of C. It is
clear that differentiation, with respect to the endpoint x, gives the relations
(41). In addition, the stream function is a constant along fluid streamlines,
or contours everywhere tangent to the local velocity vector.
For axisymmetric flow, the flux-integral analogous to (43) is
Q = 2π
∫∫
( ux RdR − uR Rdx),
(44)
S ( R ,x )
where S(R, x) is any surface bounded by the circle (R, x). The derivatives of
(44) differ from (42) by the factor 2π; hence Q = 2πψ, and we see that the
stream function is equal to (1/2π) times the flux passing through the circle
(R, x). This property ensures that the surfaces ψ = constant will define the
stream surfaces of the flow field.
As an example of the use, of the axisymmetric stream function, we consider again the flow past a Rankine ovoid, or source-sink combination, for
which the velocity potential is given by equation (33). In polar coordinates,
the potential (33) is
Figure 4.3
Contour of integration in equation (43). The fluid flux across a differential element
of this contour is u dy – v dx. A positive flux is defined as moving across C to the right,
with respect to an observer moving along C.
The Motion of an Ideal Fluid
φ( R,θ , x) = Ux −
123
m
m
[( x + a)2 + R2 ]−1/2 +
[( x − a)2 + R2 ]−1/2 .
4π
4π
(45)
From (44), we can compute the stream function as
R
R
∂φ
∂
φ R′dR′
R′ dR′ =
∂x
∂x ∫0
0
ψ ( R, x ) = ∫
{
∂ 1
m
m
Ux R2 −
[( x + a)2 + R2 ]1/2 +
[( x − a)2 + R2 ]1/2
4π
4π
∂x 2
1
m( x + a)
m( x − a)
= UR2 −
.
+
2
2 1/2
4π[( x − a)2 + R2 ]1/2
2
4π[( x + a) + R ]
=
}
(46)
In deriving (46), we have overlooked two contributions that cancel out.
Thus, for the stream function to remain continuous and single valued, as
a measure of the flux passing through the surface S defined in (44), this
surface must not pass through the source points at x =± a. The contribution
from ignoring this restriction is equal to the flux of these sources, ± m; this
is canceled by the ignored contribution to the integral in (46) from the
lower limit at R = 0.
The first term in (46) gives the stream function of the axial stream; the
second and third terms give the stream functions of a source at x = −a and
a sink at x = + a, respectively. For |x| > a, the axis R = 0 corresponds to the
streamline ψ = 0, and since the stagnation points are included on this
line, it follows that ψ = 0 is the equation of the dividing streamlines that
pass around the body surface. Hence the equation of the body surface is
given by
0=
2πU R2
( x + a)
( x − a)
−
.
+
m
[( x + a)2 + R2 ]1/2 [( x − a)2 + R2 ]1/2
(47)
In particular, setting x = 0 gives equation (36) for the maximum radius of
the body, and it is apparent that the stream function was used implicitly
in (36).
4.7 The Complex Potential
In the case of plane two-dimensional flows, analytic functions of a complex
variable can be used to represent the velocity field. This allows us to exploit
a wide variety of mathematical tools.
124
Chapter 4
Let us assume that the flow depends only on the coordinates x and y,
which are taken to be the real and imaginary parts of the complex variable
z = x + iy. We define the complex potential F(z) to be
F( z ) = φ + iψ ,
(48)
where ϕ is the velocity potential and ψ the stream function. Since the two
velocity components can be determined by differentiating either of these
real functions, it follows that
u = ∂φ / ∂x = ∂ψ / ∂y ,
v = ∂φ / ∂y = −∂ψ / ∂x,
or the real and imaginary parts of F satisfy the Cauchy-Riemann equations.
Therefore the complex potential F is an analytic function of the complex
variable z, and its derivative is
dF
= u − iv.
dz
(49)
Conversely, any analytic function defines a complex potential, and its
derivative is a plausible velocity field. Some simple analytic functions with
corresponding velocity fields are listed in table 4.1.
An example of particular interest in two dimensions is the corner flow
F(z) = zn, which gives the flow interior to a sector of included angle π/n. This
can be verified by using polar coordinates, z = reiθ, and writing the complex potential as F(z) = rneinθ. Since the imaginary part is zero on θ = 0 and
θ = π/n, these are streamlines of the flow. Special cases of particular interest are n = 2, or the flow interior to a right angle, and n = 1/2 which gives
the exterior flow around the edge of a semi-infinite flat plate. Both cases
are illustrated in figure 4.4. A significant difference between these is the
local behavior near the origin. The case n = 2 has a stagnation point. This is
Table 4.1
Simple Examples of Complex Potentials and the Corresponding Velocity Fields
F(z)
u − iv
Type of Flow
Uz
U
Stream
log z
1/z
Source
2
1/z
−1/z
Dipole
Uz + A/z
U − A/z2
Stream of velocity U flowing past a circle of radius (A/U)1/2
zn
n zn−1
Corner flow within a sector of angle π/n.
The Motion of an Ideal Fluid
125
Figure 4.4
Simple complex potentials and the corresponding streamlines.
characteristic of interior corner flows, where the included angle is less than
180°, or where the exponent n is greater than one. On the other hand, for
n = 1/2 an infinite velocity occurs at the sharp edge, and this is characteristic of all exterior corner flows where 1/2 < n < 1.
4.8 Conformal Mapping
A particularly useful facet of the application of analytic function theory to
two-dimensional flows is the use of conformal mapping techniques. Here,
the objective is to map a complicated geometry onto a simpler one, replacing the fluid flow and body profile in the physical z-plane by a fictitious
flow past a hypothetical profile in, say, the ζ-plane, where ζ is related to z
by a known analytic function.
One of the simplest examples is the corner flow considered in the preceding section. A corner with included angle θ can be mapped onto a straight
line in the ζ-plane by the transformation ζ = zπ/θ, as shown in figure 4.5.
In the ζ-plane the flow past the straight line is simply a uniform stream;
hence the complex potential is F(ζ) = ζ. Inverting the mapping procedure,
it follows that
F( z ) = z π /θ .
(50)
126
Chapter 4
Figure 4.5
The complex potential F(z) = zπ/θ, and the corresponding conformal mapping onto
the plane ζ = zπ/θ.
This answer agrees with the result shown in table 4.1, which was deduced
by the indirect process of first choosing a potential and subsequently determining the flow geometry.
As a more interesting example of conformal mapping, let us generalize
the complex potential for the flow past a circle. Replacing the potential by
the variable ζ we are led to consider the mapping
A − iθ
e
r
= (Ur + A / r )cos θ + i(Ur − A / r )sin θ ,
ζ = Uz + A / z = Ure iθ +
(51)
where z = reiθ has been substituted. Equation (51) defines a family of confocal ellipses in the ζ-plane; the major semiaxis Ur + A/r and the minor
semiaxis Ur − A/r coincide with the real and imaginary axes respectively. In
particular, the circle r = (A/U)1/2 in the z-plane is mapped onto the segment
between ± 2(AU)1/2 in the ζ-plane. For convenience we shall set A = U = 1/2,
so that the circle of unit radius in the z-plane is mapped onto the straight
line (−1, 1) in the ζ-plane. Physically we can suppose that this segment corresponds with a flat plate, which is the conformal mapping of the circle. In
this special case (51) reduces to the mapping
ζ=
1
( z + 1 / z ).
2
(52)
The Motion of an Ideal Fluid
127
Solving the resulting quadratic equation for z gives
z = ζ + (ζ 2 − 1)1/2 .
(53)
Here the branch of the square root is chosen so that (ζ2 − 1)1/2 ≃ ζ as
ζ → ∞; hence z ≃ 2ζ which is consistent with the corresponding limit of
(52) as z → ∞.
With the above preliminaries we now are ready to move back and forth
between the z-plane and ζ-plane and to relate the flow past the unit circle to
the flow past a flat plate. The simplest solution to start with is the complex
potential F(ζ) = ζ, corresponding to a uniform stream in the ζ-plane parallel to the plate. Since the plate does not disturb the flow in this particular
case, the solution is complete. Transforming this complex potential back to
the z-plane by means of (53) gives the potential for the flow past the unit
circle:
F( z ) = ζ =
1
( z + 1 / z ).
2
(54)
Note that the stream velocity at infinity in the z-plane is parallel to the
x-axis, of magnitude 1/2. Thus the conformal mapping has enabled us
to recover the solution for the flow past a circle in terms of the simple
uniform-stream solution past a flat plate at zero incidence.
To proceed further, we rotate the stream velocity by 90 degrees. In the
z-plane the situation is unchanged, except that the flow is now vertical
past the circle. But in the ζ-plane the flow is normal to the flat plate. At
this point the latter solution is unknown, but the solution in the z-plane is
known, and this can be mapped into the ζ-plane. First, if z is replaced by iz,
the complex potential (54) becomes
F( z ) =
1
i( z − 1 / z ).
2
(55)
Note that, from (49), the streaming flow at infinity is positive downward
with magnitude equal to 1/2. Transforming to the ζ-plane requires that (53)
be substituted in (55). Thus the complex potential for flow normal to the
flat plate is given by
1
1
i[ζ + (ζ 2 − 1)1/2 ] − i[ζ + (ζ 2 − 1)1/2 ]−1
2
2
= i(ζ 2 − 1)1/2 .
F(ζ ) =
(56)
128
Chapter 4
In the ζ-plane the complex velocity is F'(ζ), and at infinity (56) gives a
downward streaming velocity of unit magnitude past the horizontal plate.
The complex potential can be multiplied by any real constant to change
the value of the stream velocity; for the general case of flow at normal incidence with stream velocity U, past a flat plate of width 2a, the appropriate
generalization of (56) is
F(ζ ) = iU (ζ 2 − a2 )1/2 .
(57)
Differentiating (57) gives the complex velocity
u − iv =
iU ζ
.
(ζ 2 − a2 )1/2
(58)
As in the case of the flow around a semi-infinite flat plate, the velocity is
infinite at the sharp edges with a square-root singularity. More generally
one can expect the singular behavior of the flow near a sharp corner to
depend only on the local shape near the corner.
Since the general conformal mapping (51) maps concentric circles in the
z-plane onto confocal ellipses in the ζ-plane, the flow past an ellipse can be
obtained in a similar manner.
Certain fundamental characteristics of conformal mapping transformations that we have taken for granted should be stated explicitly. The following three properties apply to any conformal transformation given by an
analytic function:
1.
The mapping is one-to-one, so that to each point in the physical
domain, there is one and only one corresponding point in the mapped
domain.
2.
Closed curves map to closed curves.
3.
Angles are preserved between the intersections of any two lines in the
physical domain and in the mapped domain.
The last property obviously is violated in the case of the flat plate, at the
two ends of the plate, because the mapping function is not analytic at these
points. This feature is exploited in the Schwarz-Christoffel transformation,
where polygonal profiles of arbitrary form are mapped onto a straight line,
utilizing products of corner mappings analogous to (50). An extensive treatment of conformal mapping is given by Milne-Thomson (1968).
Finally, the Riemann mapping theorem, which states that an arbitrary
closed profile can be mapped onto the unit circle, enables the solution for
The Motion of an Ideal Fluid
129
the flow past any practical two-dimensional body to be obtained by conformal transformation. However, the practical significance of the Riemann
mapping theorem is diminished by the numerical complexity of mapping
arbitrary body profiles, and generally speaking other techniques are equally
useful under these circumstances. Principal among the alternatives is the
representation of the body by a distribution of singularities on the body
surface. Before discussing this approach, however, we will first discuss a
three-dimensional method more akin to conformal mapping.
4.9 Separation of Variables
Aside from guessing solutions indirectly, the simplest technique for solving three-dimensional problems is separation of variables. Laplace’s equation
is separable in thirteen coordinate systems including rectangular, circular
cylindrical, elliptic cylindrical, parabolic cylindrical, spherical, conical,
parabolic, prolate spheroidal, oblate spheroidal, ellipsoidal, paraboloidal,
bispherical, and toroidal coordinates. The features of these are given in
detail by Morse and Feshbach (1953). The cylindrical coordinate systems
are generalizations from two dimensions where there is a close relation
with complex-variable methods. The essential idea is to assume a solution
for the velocity potential as the product of three functions, each depending separately on one of the three coordinates. If the coordinate system is
separable, Laplace’s equation will reduce to a system of three ordinary differential equations for these three functions.
To illustrate this method, let us return to the problem of a uniform
stream flowing past a sphere. After the introduction of spherical coordinates (r, θ, α), which are related to the Cartesian coordinates by
x = r cos θ ,
y = r sin θ cos α ,
z = r sin θ sin α ,
Laplace’s equation takes the form
1 ∂  2 ∂φ 
1
1
∂ 
∂φ 
∂ 2φ
r
+
sin
θ
+
= 0.
r 2 ∂r  ∂r  r 2 sin θ ∂θ 
∂θ  r 2 sin2 θ ∂α 2
(59)
Assume that the uniform stream at infinity is in the positive x-direction and
that the sphere is centered about the origin; then the velocity potential will
130
Chapter 4
be independent of the coordinate α. The resulting boundary-value problem
for ϕ(r, θ) is stated as follows:
1 ∂  2 ∂φ 
1
∂ 
∂φ
r
+
sin θ  = 0,
r 2 ∂r  ∂r  r 2 sin θ ∂θ 
∂θ 
∂φ / ∂r = 0 ,
r > r0 ,
on r = r0 ,
φ → Ur cos θ ,
as r → ∞.
(60)
(61)
(62)
Here r = r0 is the radius of the sphere, (60) states that the velocity potential
satisfies the Laplace equation in the fluid domain, (61) is the kinematic
boundary condition on the sphere, and (62) states that the flow tends to a
uniform stream at large radial distances from the sphere.
This problem is solved, by the method of separation of variables, by
assuming a solution of (60–62) in the form
φ( r ,θ ) = R( r )Θ (θ ).
(63)
Then, from (60),
r 2φ −1∇2φ =
1 d 2
1
d
( r R′ ) +
(sin θΘ ′ ) = 0,
R dr
Θ sin θ dθ
(64)
where a prime denotes the derivative with respect to the appropriate variable. The first term on the right side of (64) depends only on r, and the
second only on θ; if the two are to be equal and opposite for all values of
r and θ, they must each be a constant. Thus R and Θ satisfy the ordinary
second-order differential equations
d 2
( r R′ ) − RC = 0,
dr
(65)
d
(sin θΘ ′ ) + CΘ sin θ = 0.
dθ
(66)
Solutions of the second equation are the Legendre functions
Pn (cos θ ),
where C = n( n + 1),
and the Legendre functions of the second kind Qn(cos θ). If we require the
solution to be regular for 0 ≤ θ ≤ π, only the first functions can be allowed,
and n must then be zero or a positive integer. The first equation now takes
the form
The Motion of an Ideal Fluid
131
d 2
( r R′ ) − Rn( n + 1) = 0,
dr
(67)
and solutions are clearly given by
R = rn,
R = 1 / r n +1 .
Thus, with An and Bn a set of undetermined constants, the most general
solution for the velocity potential ϕ(r, θ) or (60) is
∞
φ = ∑ Pn (cos θ )[ An r n + Bn / r n +1 ].
(68)
n=0
The Legendre functions Pn(x), with n an integer, are polynomials in x; the
first three of these are
P0 ( x) = 1,
P1( x) = x,
P2 ( x) =
1
(3x2 − 1).
2
These polynomials are orthogonal in the interval −1 < x < + 1, and the term
with n = 1 in (68) is the only member of this series that can be proportional
to cos θ as required by the condition (62) at infinity. Thus, the remaining
coefficients for n > 1 must vanish in (68). The solution of (60–62) is therefore given by the potential
1
φ = U cos θ  r + r0 3 r 2  .


2
(69)
This potential is recognized as the superposition of a dipole and a uniform
stream; cf. equation (38).
The term in (68) proportional to A0 is a constant, and in (69) this has
been set equal to zero. The corresponding term B0/r is a source, which is
ruled out here because the sphere has constant volume. An alternative
problem would be the pulsation of a spherical bubble of gas within the
fluid, where the source term would be the appropriate solution.
For polar coordinates (r, θ) in two dimensions, the analog to (68) is the
expansion
∞
φ = ∑ cos nθ[ An r n + Bn / r n ] + A0 + B0 log r
(70)
n =1
together with a similar series with cos nθ replaced by sin nθ. This is equivalent to expressing the complex potential in its most general form as a
Laurent series plus a logarithmic source term.
132
Chapter 4
A common feature of (68) and (70) is that two sets of independent
radial functions arise, one singular at the origin and the other at infinity. The latter is appropriate for internal flows, where the fluid is contained
within a spherical or circular container with prescribed normal velocity
on the boundary. However, for external flows in a fluid domain that extends
to infinity radially, the solutions of Laplace’s equation proportional to rn
are ruled out, with the exception of a uniform free stream obtained when
n = 1. On the other hand, for external flows the entire set of solutions singular at r = 0 is admissible, since the fluid domain is exterior to this point.
Individual terms in this set are known as multipoles; the source and dipole
are the first two members. (The next two are quadrupoles and octapoles.) The
multipole of order n can be derived by differentiating the source potential
n times.
The method of separation of variables can be applied, using the same
scheme with more complicated coordinate systems, to determine the flow
past spheroids and ellipsoids and more generally to solve problems where
the boundary geometry coincides with one of the coordinate surfaces of the
separable systems listed above. Details of these problems and the resulting
solutions are discussed by Lamb (1932) and by Morse and Feshbach (1953).
The solutions obtained are analogous in form to (68) and (70), with undetermined coefficients that must be found from the boundary conditions
of the problem, but with more complicated special functions representing
the spatial dependence. Several applications of this technique in the field
of ship hydrodynamics are included in the collected works of Havelock
(1963).
4.10 Fixed Bodies and Moving Bodies
In the previous sections, streaming flows have been considered where the
body is fixed and the fluid at infinity has a uniform constant velocity.
Except for the change in reference frame, these are identical to corresponding problems where the body moves with constant velocity through otherwise undisturbed fluid, provided the relative velocity between the body
and the fluid at infinity is the same in both cases. However, the boundary
conditions must be changed if the fluid velocity is referred consistently to
a fixed reference frame, and the two solutions will differ by the free-stream
potential.
The Motion of an Ideal Fluid
133
Table 4.2
Comparison of Flows for Circle and Sphere of Radius r0
Horizontal Streaming Flow Past a Fixed Body
Circle
Sphere
Boundary Condition on Body
∂ϕ/∂r = 0
∂ϕ/∂r = 0
Boundary Condition at Infinity
ϕ → Ux = Ur cos θ
Velocity Potential
2
ϕ = U(r + r0 /r) cos θ
ϕ → Ux = Ur cos θ
φ = U ( r + 21 r0 3 /r 2 )cos θ
Horizontal Translation of Body with Fluid at Rest at Infinity
Circle
Sphere
Boundary Condition on Body
∂ϕ/∂r = U cos θ
∂ϕ/∂r = U cos θ
Boundary Condition at Infinity
ϕ→0
ϕ→0
Velocity Potential
ϕ = −(Ur02/r) cos θ
φ = −( 21 Ur0 3 /r 2 )cos θ
To illustrate this comparison, table 4.2 shows the boundary conditions
and velocity potentials for a circle and a sphere when the fluid streams
past the fixed body, as well as when the fluid is at rest with the body moving. The stream velocity in the former case is + U in the + x-direction; the
body velocity in the second case is denoted by the same symbol, with the
convention that a positive velocity of the body is in the + x-direction. Thus
the relative velocities differ in sign, and for this reason the dipole moments
of the corresponding solutions are equal in magnitude, but of opposite
signs.
When the body moves with an unsteady motion, the same kinematic
conclusions apply to the velocities of the body and fluid, with U(t) a function of time. In this instance, however, the problems in each frame of
reference will differ from the dynamic standpoint; this difference will be
examined in section 4.17.
4.11 Green’s Theorem and Distributions of Singularities
Earlier we showed that certain bodies could be represented by suitable
combinations of a source and sink or by a point dipole. Given the effect
of these singularities on the surrounding fluid, it seems plausible that a
larger number of them could be used to represent more complicated body
shapes. In the most general case, one might envisage a continuous volume
distribution of singularities within the body, extending out to its surface.
134
Chapter 4
Alternatively, from the standpoint of the flow exterior to the body, a distribution on the body surface alone might suffice. This latter speculative
conjecture turns out to be correct, and in the process of establishing it a
number of useful results will be obtained.
For reasons that will become clear later, let us consider two solutions
of Laplace’s equation in a volume  of fluid bounded by a closed surface
S. Denoting these two potentials by ϕ and φ and applying the divergence
theorem we get
 ∂ϕ
∂φ 
∫∫ φ ∂n − ϕ ∂n  dS = ∫∫∫ ∇ ⋅ (φ∇ϕ − ϕ∇φ )d
S

= ∫∫∫ (φ∇2ϕ + ∇φ ⋅ ∇ϕ − ϕ∇2φ − ∇ϕ ⋅ ∇φ )d
(71)

= 0.
This important result is a form of Green’s theorem that will be utilized later in
a variety of contexts. Here we shall consider the consequence of replacing
φ by the potential of a source.
In the following analysis it is convenient to set the source strength
m = −1. Of more significance is the position of this unit source, which
must be carefully specified. We shall define the source point ξ ≡ (ξ, η, ζ) as the
position of the source in the coordinates x = (x, y, z). For a unit source the
potential at the field point x is given by
ϕ = 1 / 4π r = (1 / 4π )[( x − ξ )2 + ( y − η )2 + ( z − ζ )2 ]−1/2 .
(72)
In accordance with the principle of reciprocity, the value of (72) is
unchanged if the source point and field point are interchanged. By the
same token, (72) is a solution of Laplace’s equation with respect to ξ as well
as x. Thus, in utilizing this source potential in Green’s theorem (71) we
can integrate with respect to either coordinate system. Mathematically the
results are equivalent, but different physical interpretations follow.
In the subsequent derivation we shall perform the surface integration of
(71) with respect to the coordinates of the source point ξ. This requires that
the potential ϕ and the normal derivative ∂ϕ/∂n be defined with respect to
(ξ, η, ζ) by a simple change of the dummy variable of integration. Physically we shall integrate over a continuous distribution of sources and normal dipoles that are located on the surface S, with a fixed value of the field
point x.
The Motion of an Ideal Fluid
135
Figure 4.6
Surfaces of integration for Green’s theorem. In (a) the field point is interior to S,
surrounded by a small spherical surface Sε; in (b) the field point is on the boundary
surface S and Sε is a hemisphere.
Substituting (72) in (71) requires caution, for the source potential does
not satisfy the Laplace equation at the singular point r = 0, and thus (71) is
not valid when the source point is situated within . This difficulty can be
circumvented by surrounding the source point by a small sphere of radius
r = ε with surface Sε as shown in figure 4.6 (a). Then S + Sε is a closed2 surface
surrounding the volume of fluid interior to S but exterior to Sε; within this
volume, (72) is regular. Thus, (71) can be replaced by
1
4π

∂ 1 1 ∂φ 
∫∫ φ ∂n r − r ∂n  dS = 0,
(73)
S + Sε
or
1
4π

∂ 1 1 ∂φ 
1

∂ 1 1 ∂φ 
∫∫ φ ∂n r − r ∂n  dS = − 4π ∫∫ φ ∂n r − r ∂n  dS.
S
(74)
Sε
In the limit ε → 0, the contribution from the integral over Sε in (74) can be
evaluated under the assumption that the velocity potential and its normal
derivative on Sε are both regular. The area of Sε is 4πr2, while the normal
derivative of 1/r is −∂/∂r(1/r) = 1/r2. (Note that the normal is positive when
pointing out of the fluid volume and hence in the direction opposite to r on
Sε.) Thus the first term in the integrand on the right side of (74) is singular
in proportion to 1/r2, and when it is multiplied by the area 4πr2, a finite
limit will result as ε → 0, whereas the weaker singularity of the second term
136
Chapter 4
will give no contribution. For sufficiently small ε, the potential ϕ may be
assumed constant and taken outside of the integral sign, so that the final
limiting value of the right-hand side of (74) becomes
−
1
∂ 1
φ( x, y , z )∫∫
dS = −φ( x, y , z ).
nr
4π
∂
Sε
(75)
Thus, if (x, y, z) is inside S,
φ( x, y , z ) = −
1
4π

∂ 1 1 ∂φ 
∫∫ φ ∂n r − r ∂n  dS.
(76)
S
Equation (76) can be regarded as a representation of the velocity potential in terms of a normal dipole distribution of moment ϕ and a source distribution of strength − ∂ϕ/∂n distributed over the boundary surface S. Thus,
in general, the flow can be represented by a suitable distribution of dipoles
and sources on the boundary surface S. By further manipulation with
Green’s theorem, one can obtain alternative integrals involving sources or
dipoles only, as shown by Lamb (1932). (See problem 13.)
If the point (x, y, z) is situated on the surface S, the surface Sε is chosen to
be a small hemisphere that indents the original surface S inside the source
point, as in figure 4.6(b). The contribution from this hemisphere is just half
that given by (75), so that (76) is replaced by
φ( x, y , z ) = −
1
2π

∂ 1 1 ∂φ 
∫∫ φ ∂n r − r ∂n  dS.
(77)
S
Here, the surface integral must be defined to exclude the immediate vicinity
of the singular point, i.e., the locally-plane infinitesimal surface bounded
by the intersection of S and the hemisphere Sε. This situation is analogous
to a principal value integral, except that in (77) the precise shape of the
excluded infinitesimal area is not important.
Finally, when the point (x, y, z) is situated outside S, (71) is valid without
modification and the left-hand side of (76) or (77) is replaced by zero. Similar results hold in two dimensions, with the source potential 1/r replaced
by log r and with the surface integrals in the above relations replaced by
line integrals; in this case the factors 4π and 2π are multiplied by −1/2, corresponding to the difference in the radial flux between the singularities 1/r
and log r.
The Motion of an Ideal Fluid
137
Equation (77) is frequently used for constructing the velocity potential
due to the motion of a ship hull or other moving body. Generally, the normal derivative ∂ϕ/∂n is known on the body, so that (77) is an integral equation for the determination of the unknown potential; it may be solved by
approximations or numerical techniques.
For a body moving in an otherwise infinite and unbounded fluid, the
appropriate closed surface S enclosing the fluid volume must include both
the body surface SB and an additional control surface SC surrounding the
body. It then can be argued that SB + SC together form a closed surface surrounding the fluid volume  or, at least, that portion of  within a finite
distance of the body. The contribution from SC to the integral in (77) can
be estimated from the expansion of a general three-dimensional velocity potential in spherical harmonics, (69). For large spherical radius r, the
potential due to the presence of the body is of order r−1, and ∂ϕ/∂n of order
r−2. Thus the integrand of (77) is of order r−3, whereas on SC the differential
area dS will increase in proportion to the square of the distance from the
origin. It then follows that the contribution to (77) from SC is of order r−1,
and vanishes in the limit where this surface is an infinite distance from the
body. Therefore, for a body moving in an unbounded fluid, the integral in
(77) can be taken simply over the surface of the body SB.
In many problems, however, the body may move in a fluid bounded by
other boundaries, such as the free surface, the fluid bottom, or possibly lateral boundaries such as canal walls. In each of these cases additional boundary conditions are imposed, and there is often a computational advantage
in solving (77) if the source potential is modified to satisfy the same boundary conditions as ϕ. In this context the Green function
G( x, y , z ; ξ , η,ζ ) = 1 / r + H ( x, y , z ; ξ , η,ζ ),
(78)
where H is any function that satisfies the Laplace equation, can be substituted for the source potential in (76–77) since (71) is valid for the contribution from the regular function H. Thus, with the Green function defined by
(78), we can state that
0


 φ ∂G − G ∂φ  dS = −2πφ( x, y , z ) ,


∫∫  ∂n ∂n 
S
 −4πφ( x, y , z )


(79)
138
Chapter 4
for (x, y, z) outside, on, or inside the closed surface S. The regular function
H can be chosen to suit any additional boundary conditions that may be
imposed. If a function H can be found with the property that
∂G / ∂n = 0
(80)
on the boundary surfaces of the fluid, the unknown term in the integrand
of (79) vanishes. With this choice of the Green function, (79) provides an
explicit solution for the potential in terms of the prescribed normal velocity
on the boundaries. This Green function corresponds to the velocity potential of a source, in the presence of the appropriate fixed boundary surfaces
of the problem. Unfortunately, this source potential is not known, except
for some very simple body geometries.
One type of body geometry for which the source potential is known is
the thin nonlifting planar surface that can be associated with a symmetrical
thin hydrofoil at zero angle of attack and with a ship hull of small beam. For
these situations the body surface is to first approximation a flat disc, and
the source potential 1/r itself satisfies a condition of zero normal velocity
on the disc, provided the source is situated on the disc. Thus, thin bodies of
this type can be represented hydrodynamically by a center-plane distribution of simple sources, of strength proportional to the normal velocity on
the body surface, provided only that the flow is symmetrical with respect
to the center plane. This approximation forms the basis for two important
topics: the use of a source distribution to represent the thickness effects of
thin wings or hydrofoils, and the Michell theory for the wave resistance of
thin ships.
4.12 Hydrodynamic Pressure Forces
One of the primary reasons for studying the fluid motion past a body is
our desire to predict the forces and moments acting on the body due to
the dynamic pressure of the fluid. Thus, we wish to consider the six components of the force and moment vectors, which are represented by the
integrals of the pressure over the body surface, or
F = ∫∫ pn dS ,
(81)
M = ∫∫ p( r × n ) dS.
(82)
SB
SB
The Motion of an Ideal Fluid
139
Here, the normal vector n is taken to be positive when pointing out of the
fluid volume and hence into the body. Substituting for the dynamic pressure from Bernoulli’s equation, we get
∂φ 1
F = − ρ ∫∫  + ∇φ ⋅ ∇φ  n dS ,
∂t 2


SB
(83)
∂φ 1
M = − ρ ∫∫  + ∇φ ⋅ ∇φ  ( r × n )dS.
∂t 2

SB 
(84)
Equations (83) and (84) can be recast by using Gauss’ theorem,
∫∫∫ ∇f d = ∫∫ fn dS,
(85)
S

together with the transport theorem in equation (3.11). First let us surround the body by a fixed control surface SC, exterior to the body surface
SB as shown in figure 4.7. Thus SB + SC forms a closed surface, enclosing
a fluid volume (t ). The rate of change of the fluid momentum in this
volume is
ρ
d
dt
∫∫
φ n dS = ρ
SB + SC
d
∇φ d 
dt ∫∫∫
 (t )
∂φ
= ρ ∫∫∫ ∇   d  + ρ ∫∫ ∇φ(U
U ⋅ n )dS
∂t 

SB + SC
 (t )
=ρ
∫∫
SB + SC
(86)
 ∂φ n + ∇φ(U ⋅ n ) dS ,
 ∂t

where first Gauss’ theorem and then the transport theorem have been used.
Figure 4.7
Fixed control surface SC surrounding the moving body surface SB.
140
Chapter 4
On the fixed control surface SC, U·n = 0, and the time-derivative and
surface integration can be interchanged. As a result, the first and last surface
integrals in (86) may be equated separately over SC and hence over SB. Using
the boundary condition ∂ϕ/∂n = U·n on the body surface gives
ρ
d
∂φ
∂φ
φ n dS = ρ ∫∫  n +
∇φ  dS.
∂
∂
dt ∫∫
t
n


SB
SB
(87)
Adding (87) and (83) gives the hydrodynamic pressure force in the alternative form
F = −ρ
d
dt
1


∫∫ φn dS + ρ∫∫  ∂φ / ∂n∇φ − 2 ∇φ ⋅ ∇φn dS.
SB
(88)
SB
Moreover, using the divergence and Gauss theorems,
2
 ∂  ∂φ
 1  ∂φ  
 ∂φ ∇φ − 1 ∇φ ⋅ ∇φ n  dS =
φ
−
∇
∇

∫∫  ∂n
∫∫∫  ∂xj  ∂xj  2  ∂xj   d

2
SB + SC
 (t ) 

∂ 2φ
= ∫∫∫ ∇φ
d  = ∫∫∫ ∇φ∇2φ d  = 0.
∂x j ∂x j
 (t )
 (t )
(89)
It follows that the second integral in (88) may be replaced by the negative
of the same integral over SC to give the desired expression for the force
F = −ρ
d
dt
 ∂φ
1

∫∫ φn dS − ρ∫∫  ∂n ∇φ − n 2 ∇φ ⋅ ∇φ  dS.
SB
(90)
SC
A physical interpretation of this result can be inferred from the fact
that the second member of (86) is the rate of change of fluid momentum
enclosed between SB and SC, which must be equal to the sum of the external
pressure, forces acting on SB + SC plus the rate of flux of momentum across
SC. Thus
FB + FC + ρ ∫∫
SC
∂φ
d
∇φ dS = − ρ ∫∫∫ ∇φ d 
∂n
dt  ( t )
= −ρ
d
∂φ
φ n dS − ρ ∫∫ n dS.
dt ∫∫
∂t
SB
SC
(91)
If Bernoulli’s equation is used as in (83) to evaluate FC, (90) follows directly.
Similarly, the relation
The Motion of an Ideal Fluid
141
∫∫∫ ∇ × Q d = ∫∫ (n × Q ) dS
(92)
can be used to derive from (84) an alternative expression for the moment
in the form
M = −ρ
d
∂φ
1
φ( r × n ) dS − ρ ∫∫ r ×  ∇φ − ∇φ ⋅ ∇φ n  dS.
∂
dt ∫∫
n
2


SB
SC
(93)
Equations (90) and (93) express the hydrodynamic pressure force and
moment acting on the body surface SB in an alternative form to the direct
pressure integrals (83–84). By comparison with the latter, (90) and (93)
involve a simpler integral over the body surface but an additional integral
over the control surface surrounding the body. At first glance, the advantage of the new equations is not apparent—indeed they might appear more
complicated. In practice, however, the flexibility afforded by the freedom
to choose the control surface in an optimum manner often simplifies the
computation of the force and moment. This is the principal justification for
the analysis required to derive (90) and (93).
By virtue of (89), the control surface SC could be placed within the body,
in deriving (90) from (88), and similarly in (93). The only restriction here
is that SC must remain exterior to any singularities within the body, but it
may be arbitrarily close to these singularities. On this basis one can derive
the Lagally theorem for the force and moment, in terms of products of the
singularity strengths and gradients of the velocity potential at the points of
location of the singularities. Details are given by Cummins (1957), and by
Landweber and Yih (1956), and reviewed by Milne-Thomson (1968).
Conversely, the evaluation of the integrals over SC may be simplified by
removing this control surface to the far field, where the details of the flow
past the body are unimportant. This procedure will be followed in the next
section, where we analyze the case of a rigid body moving in an otherwise
infinite and unbounded fluid. For this particular problem, the integrals
over SC are simplified to the ultimate extent where they vanish!
4.13 Force on a Moving Body in an Unbounded Fluid
Now let us consider the hydrodynamic force and moment acting on a rigid
body, which moves in an arbitrary manner in an otherwise unbounded
fluid. In this case the contributions from the integrals over the control
142
Chapter 4
surface SC, in (90) and (93), can be estimated in an analogous manner to
that employed in connection with the corresponding integral in Green’s
theorem. In particular, for large spherical radius r, the potential3 due to the
body motions will tend to zero at a rate proportional to r−2, and the gradient
will vanish as r−3. Let SC be a sphere of large radius; since the surface element
dS is proportional to r2, the contribution from the integrals over SC in (90)
and (93) will be of order r−4 and r−3, respectively. As r tends to infinity, these
integrals will vanish, and the force and moment are given by the simple
expressions
F = −ρ
d
dt
M = −ρ
∫∫ φn dS,
(94)
SB
d
dt
∫∫ φ(r × n) dS.
(95)
SB
These results imply that the total momentum of the fluid is equal to ρ times
the integral in (94), and the moment of momentum is similarly proportional to the integral in (95).
In the special but important case of steady translation of the body, the
integral in (94) will not depend on time, and hence we obtain D’Alembert’s
“paradox.” This states that no hydrodynamic force acts on a body moving
with constant translational velocity in an infinite, inviscid, and irrotational
fluid. A moment may exist in this case, however. To verify this, we note that
problems of steady translation are independent of time only when viewed
in a reference frame moving with the body. If the body moves with velocity
U1 parallel to the x1-axis, the appropriate moving coordinate system is
x1′ = x1 − U 1t ,
x2′ = x2 ,
x3′ = x3 ;
the potential
φ ′( x1′ , x2′ , x3′ ) ≡ φ( x1′ + U 1t , x2′ , x3′ , t )
(96)
will be independent of time. On the body surface SB, the unit normal also
will be independent of time, but not the cross-product
r × n = ( x2 n3 − x3n2 , x3n1 − x1n3 , x1n2 − x2 n1 )
= r ′ × n + ( 0, −U 1tn3 ,U 1tn2 ).
More generally, if U is the body velocity,
(97)
The Motion of an Ideal Fluid
143
r × n = r ′ × n + t(U × n ).
(98)
Hence, for steady translation, the moment (95) is given by
M = − ρ ∫∫ φ(U × n ) dS = − ρU × ∫∫ φ n dS ,
SB
(99)
SB
and it is perpendicular to the velocity vector of the body. This moment is
equal to zero if the body is symmetrical with respect to the two directions
normal to U. Thus, for example, a submarine in steady forward motion
with port-starboard symmetry may experience only a pitching moment
(bow up or bow down).
Now let us consider the most general case—unsteady motion with six
degrees of freedom. If the translational velocity is U(t) and the body is rotating with angular velocity Ω(t) about an origin that moves with the body,
the velocity potential must satisfy the boundary condition
∂φ
= U ⋅ n + Ω ⋅ (r ′ × n)
∂n
(100)
on the body surface. Here r' is the radius vector from the center of rotation.
It is convenient to define the six velocity components by the redundant
notation
U = (U 1 ,U 2 ,U 3 ),
W = ( Ω1 , Ω2 , Ω3 ) ≡ (U 4 ,U 5 ,U 6 ).
Thus, U1, U2, and U3 denote the three components of translational velocity
(surge, heave, and sway), and Ω1 = U4, Ω2 = U5, and Ω3 = U6 denote the corresponding rotational velocity components (roll, yaw, and pitch).
The boundary condition (100) suggests that the total potential be
expressed as the sum
φ = U iφi .
(101)
Here the summation convention applies, with i = 1, 2, …, 6. Physically,
each ϕi represents the velocity potential due to a body motion with unit
velocity in the ith mode. Note that the dimensions of ϕ are L2T−1, whereas
those of U and Ω are LT−1 and T−1, respectively. Therefore the potentials ϕi in
(101) are not dimensionally homogeneous; the units are L and L2, respectively, for i = 1, 2, 3, and i = 4, 5, 6.
144
Chapter 4
The potential defined by (101) satisfies the body boundary condition
(100), provided each component ϕi satisfies the corresponding condition
∂φ i
= ni ,
∂n
i = 1, 2, 3,
(102)
∂φ i
= ( r ′ × n )i −3 , i = 4, 5, 6.
∂n
(103)
The boundary-value problem for each ϕi is completed by the usual requirements that ∇2φi = 0 in the fluid volume and ϕi = O(r−2), or smaller, as r → ∞.
These six potentials, when expressed in body-fixed coordinates, will depend
only on the body geometry via the boundary conditions (102) and (103),
and do not depend on the velocities Ui or time. Thus, the linear decomposition (101) isolates the time dependence of the velocity potential, and
substitution in (94) yields the hydrodynamic force expression
F = −ρ
d
U i (t )∫∫ φi ( x1′ , x2′ , x3′ )n dS.
dt
SB
(104)
This last integral will depend on time, because of the effect of body rotation
on the vector n. From vector analysis this contribution is
dn
= W × n,
dt
(105)
since Ω is the rotational velocity vector of the body-fixed coordinates. From
(104) it follows that
F = − ρU i ∫∫ φi n dS − ρU i Ω × ∫∫ φi n dS.
SB
(106)
SB
The corresponding result for the moment can be derived from (95) after
decomposition of the position vector r in the form
r = r0 (t ) + r ′.
(107)
Here r0 is the vector from the origin of the fixed reference system to the
point r' = 0 which moves with the body, and thus dr0/dt = U. On the other
hand, (r' × n) is a vector fixed with respect to the body, and thus its timederivative is Ω × (r' × n) as in (105). Substituting (107) in (95) and evaluating
the time derivative, the moment can be expressed in the form
The Motion of an Ideal Fluid
145
M = r0 × F − ρU iU × ∫∫ φi n dS − ρU i ∫∫ φi ( r ′ × n ) dS
SB
SB
− ρU i W × ∫∫ φi ( r ′ × n ) dS.
(108)
SB
The first term in (108) is the moment due to the force F acting at the body
origin r' = 0; therefore, the remaining terms in this expression give the
moment about the same point.
At this stage the time derivatives in (94–95) have been evaluated with
respect to the space-fixed reference frame r. This reference frame loses its
significance if the force and moment are redefined with respect to the bodyfixed coordinates. In fact, the need to discriminate between these two coordinate systems can be avoided simply by setting r0 = 0, with the assumption
that the two coordinate systems coincide at the particular instant of time
under consideration. However, this artifice is not essential, and we shall
proceed hereafter without assumption as to the origin of the space-fixed
coordinates.
An indicial notation will be adopted for the components of the force
and moment, with respect to the body-fixed reference frame. These will be
denoted by Fj and Mj respectively. The subsequent representation of crossproducts is facilitated by using the alternating tensor εjkl, which is equal to +1
if the indices are in cyclic order (123, 231, 312), equal to −1 if the indices
are acyclic (132, 213, 321), and equal to zero if any pair of the indices are
equal. With this notation, the jth component of a vector cross-product can
be written in the indicial notation as
( A × B) j = ε jkl Ak Bl ,
(109)
where the summation convention is implied for k and l.
With the conventions noted above (106) may be rewritten in the indicial
form
Fj = − ρU i ∫∫ φi n j dS − ρε jklU j Ω k ∫∫ φi nl dS.
SB
(110)
SB
Similarly, from (108), the components of the moment about the body-fixed
origin are given by
M j = − ρU i ∫∫ φi ( r ′ × n ) j dS − ρε jklU i Ω k ∫∫ φi ( r ′ × n )l dS
SB
SB
− ρε jklU iU k ∫∫ φi nl dS.
SB
(111)
146
Chapter 4
It is apparent that the force and moment depend on the body shape and
on the potentials ϕi only in terms of the integrals shown in (110–111); in
view of the boundary conditions (102–103), these integrals can be written
in the alternative forms
∫∫ φ n dS = ∫∫ φ
i
j
SB
SB
i
∂φ j
dS ,
∂n
∫∫ φ (r ′ × n) dS = ∫∫ φ
i
j
SB
SB
i
∂φ j + 3
dS.
∂n
(112)
(113)
The importance of these quantities in (110–111) suggests the definition of
the added-mass tensor
mji = ρ ∫∫ φi
SB
∂φ j
dS
∂n
(114)
since, with this substitution in (110–111), the latter equations can be
expressed in the form
Fj = −U i mji − ε jklU i Ω k mli ,
(115)
M j = −U i mj + 3,i − ε jklU i Ω k ml + 3,i − ε jklU iU k mli .
(116)
Here the three indices j, k, l take on the values 1, 2, 3, whereas the index i is
used to denote the six components of the velocity potential in accordance
with (101).
These expressions (115–116) for the force and moment depend only on
the body velocity and acceleration components and the added-mass coefficients mij. No surface integrals are involved, other than those in (114) which
define the added-mass coefficients; so from a computational standpoint,
(115) and (116) are much simpler than the pressure integrals (81) and (82).
Indeed, this is the principal justification for the extensive analysis just performed. The added-mass coefficients depend only on the body geometry;
they can be regarded as the most important hydrodynamic characteristics
of the body, except in the case of steady translation where (115) predicts no
force and the viscous drag is clearly more important.
It may be worthwhile to review briefly the assumptions implicit in
(115–116). First, the fluid is ideal and irrotational, and viscous forces are
neglected. Second, the body has been taken to be rigid and of constant
volume; its motions are defined by three translational and three rotational
The Motion of an Ideal Fluid
147
velocity components. Third, except for the presence of the body itself, the
fluid is unbounded and of infinite extent.
For a variety of problems in marine hydrodynamics, the second assumption is the most acceptable of the three. We have shown that the assumption of irrotational flow is consistent with the assumption of an ideal fluid
motion; however, important modifications are required for lifting surfaces
with sharp trailing edges and associated shed vorticity. The relative importance of the inertial forces and moments (115–116) by comparison to viscous forces and moments is discussed in section 2.11. Finally, with respect
to the third assumption, (115–116) will have to be modified in cases where
the fluid domain is bounded by other boundaries. The presence of a free
surface is the most important and troublesome of these possibilities, but
simpler problems also occur, as, for example, when the body moves in
proximity to a plane rigid wall. The latter situation can be dealt with by the
method of images, an approach which will be described briefly in the final
section of this chapter.
4.14 General Properties of the Added-Mass Coefficients
As the name implies, there is an important analogy of the added-mass coefficients mij with the body mass and moments of inertia. In section 4.16 we
shall demonstrate this analogy by showing that the force and moment due
to the body mass are of precisely the same form as (115–116). Thus from
the physical standpoint the added-mass coefficients represent the amount
of fluid accelerated with the body. However, the added-mass coefficients for
translation generally differ depending on the direction of the body motion,
as opposed to Newton’s equation F = ma where the body mass m is independent of the direction of the acceleration a. Also, in the absence of body
symmetry, the cross-coupling coefficients such as m12, m13, and m23 are nonzero, implying that the hydrodynamic force differs in direction from the
acceleration.
The added mass can be interpreted as a particular volume of fluid particles that are accelerated with the body. Strictly, however, the particles
of fluid adjacent to the body will accelerate to varying degrees, depending on their position relative to the body. In principle, every fluid particle
will accelerate to some extent as the body moves, and the added mass is a
weighted integration of this entire mass.
148
Chapter 4
One convenient feature of the added-mass coefficients is their symmetry, mij = mji. To confirm this property, Green’s theorem (71) is applied to
the potentials ϕi and ϕj over the closed surface SB + SC. The contribution
from the external surface SC will vanish as r → ∞, as noted above. Hence, it
follows that

∫∫ φ
SB
i
∂φ j
∂φ 
− φ j i  dS = 0,
∂n
∂n 
(117)
or
mij = mji .
(118)
Thus, there are twenty-one independent coefficients, and this number is
reduced substantially if the body is symmetrical about one or more axes.
A simple relation exists between the added-mass coefficients and the
kinetic energy of the fluid. From (114) and the divergence theorem,
mij = ρ ∫∫ φ j
SB
∂φ i
dS = ρ ∫∫∫ ∇ ⋅ (φ j∇φi )d 
∂n

= ρ ∫∫∫ (∇φi ⋅ ∇φ j )d 
(119)

since ∇2φi = 0 . Here, the vanishing of ϕ at infinity has been invoked to omit
the surface integral at infinity, and the volume integrals are over the entire
fluid volume. The total velocity potential in the fluid is given by (101), and
the kinetic energy of the fluid is
1
1
ρ (∇φ ⋅ ∇φ )d  = ρ ∫∫∫ (U i∇φi ) ⋅ (U j∇φ j )d 
2 ∫∫∫
2
1
= ρU iU j ∫∫∫ ∇φi ⋅ ∇φ j d  .
2
T=
(120)
Combining this result with (119) gives
T=
1
U iU j mij .
2
(121)
Therefore the added-mass coefficients are the constants in a quadratic equation for 2T, in terms of the body velocities. This result can be obtained from
physical arguments by equating the change in fluid energy to the work
done in accelerating the body, but for rotational modes (i = 4, 5, 6) the
rate of rotation of coordinates must be accounted for, as in equations (105)
and (107).
The Motion of an Ideal Fluid
149
The third and final property of the added-mass coefficients considered
here is a relation between the translation added-mass coefficients and the
dipole moment describing the fluid motion at large distances from the
body. For sufficiently large radial distances r from the body, the disturbance
due to the body can be written in the form
φ=
1
A0
+ A ⋅ ∇   + O(1 / r 3 ).
r
r 
(122)
Here, A0 is the net source strength, A the dipole moment, and neglected
terms correspond to higher-order multipoles (quadrupoles, etc.). If the body
is rigid, there can be no net flux out of the body surface SB, and hence from
continuity the source strength A0 = 0. For symmetrical rotational modes
the dipole moment will vanish, and thus we restrict ourselves here to the
translations (i = 1, 2, 3), where it can be assumed that
φi Aij
∂ 1
,
∂x j r
as r → ∞.
(123)
Thus, the far-field disturbance is identical to that of a dipole or a combination of dipoles. We recall the case of a sphere (38) where from symmetry
Aij = 0 for i ≠ j, and where (123) is exact for all r. In effect, translation of an
arbitrary three-dimensional body will cause a far-field fluid motion qualitatively similar to a sphere, and the details of the body shape are unimportant
in the far field.
An equation for the dipole moment Aij can be obtained from Green’s
theorem. Using (76), we may write the potential at a point exterior to the
body in the form
φ( x, y , z ) =
{
∂ϕ
∂n
1
4π
∫∫
−φ
∂
[( x − ξ )2 + ( y − η )2 + ( z − ζ )2 ]−1/2 ] dS ,
∂n
SB
[( x − ξ )2 + ( y − η )2 + ( z − ζ )2 ]−1/2
}
(124)
where the first term is a distribution of sources and the second a distribution of normal dipoles. If r = [x2 + y2 + z2]1/2 is large compared to the body
dimensions and terms of order r−3 are neglected, we can write from Taylor
series expansions
[( x − ξ )2 + ( y − η )2 + ( z − ζ )2 ]−1/2 1
1
− x ⋅ ∇  
r
r 
150
Chapter 4
and
∂
1
[( x − ξ )2 + ( y − η )2 + ( z − ζ )2 ]−1/2 − n ⋅ ∇   .
∂n
r 
Thus, the potential at large distances is
φ=
1  ∂φ   1  1   ∂φ
1
−
dS
x
− φ n dS  ⋅ ∇   ,
  r 
4π  ∫∫ ∂n   r  4π  ∫∫  ∂n
(125)
which may be compared with (122). The first term vanishes if there is no
net flux through the body surface, and
φ=−

1   ∂φ
1
1
− φ n dS  ⋅ ∇   ≡ A ⋅ ∇   .
 x

n
r
r
4π  ∫∫
∂



SB

(126)
Thus, the coefficients Aij in (123) may be expressed by the integrals
Aij = −
1
4π

∫∫  x
SB
j
∂φ i
− φi n j  dS.

∂n
(127)
Moreover, recalling the body boundary conditions (102), equation (127)
may be replaced by
∂φ j 
1 
 xj ni − φi
 dS
4π ∫∫ 
∂n 
1
=
( ∀δ ij + mij / ρ ), i , j = 1, 2, 3.
4π
Aij = −
(128)
Here, ∀ is the body volume, mij the added-mass coefficient as defined by
(112), and δij is the Kroenecker delta function
0 if i ≠ j ;
δ ij = 
1 if i = j.
Thus, from equation (128), the far-field disturbance and associated dipole
moment Aij are directly related to the body volume and the added-mass
coefficients mij. From the symmetry of mij it follows that Aij = Aji. Finally,
we note without rederivation the analogous results for two-dimensional
motion,
φi Aij
and
∂
log r ,
∂x j
(129)
The Motion of an Ideal Fluid
Aij = −
1
( Sδ ij + mij / ρ ),
2π
151
(130)
where S is the cross-sectional area.
4.15 The Added Mass of Simple Forms
The simplest examples of added-mass coefficients are those for a circular
two-dimensional cylinder, and in three dimensions for a sphere. For the
circular section, the dipole moment is obtained from equation (39), and
comparison with (129) gives
Aij = − R2δ ij .
(131)
Here R is the cylinder radius, and the minus sign is inserted because the
streaming flow is positive in (39); hence the body velocity relative to the
fluid is −U. From (130) it follows that
mij = πρ R2δ ij = ρSδ ij .
(132)
Thus, the added mass of a circular cylinder is precisely equal to the displaced mass of fluid. For a sphere, equation (38) yields the dipole moment
Aij = 21 R3δ ij .
(133)
The minus sign in this equation is canceled by the minus sign associated
with the derivative in (123). From (128) it follows that
mij − 23 πρ R3δ ij = 21 ρ∀δ ij ,
(134)
so that for a sphere, the added mass is half of the displaced mass. This
reduced added-mass effect, by comparison with the two-dimensional case,
can be explained by stating that the sphere obstructs less fluid, since the
fluid can flow around the sphere in all meridional planes.
In many situations involving elongated or cylindrical bodies, such as
slender pilings, ship hulls, and mooring cables, the three-dimensional
added-mass coefficients can be approximated by a strip theory synthesis, in
which the flow at each section is assumed to be locally two dimensional.
Thus, the simpler two-dimensional added-mass coefficients are of practical importance. The case of a circular cylinder is given by (132), and the
corresponding results for an elliptical cylinder can be derived in a similar
manner, taking advantage of the conformal transformation (51) to map the
152
Chapter 4
ellipse onto a circle. The added-mass coefficients of an ellipse are closely
related to those of a circle, as shown in table 4.3. For acceleration parallel
to either of the principal axes of the ellipse, the added mass is equal to that
of a circle having the same radius as the semiaxis normal to the motion.
The same results hold for a flat plate, or an ellipse of vanishing minor axis,
where the added mass for motion normal to the plate is equal to the displaced mass of the circumscribed circle.
The last result can be confirmed from the complex potential (57), either
by direct integration over the plate using (114) or by computing the effective dipole moment and using (130). For acceleration in the plane of the
plate there is no added-mass force, in the absence of viscous effects. Similarly, the added moment of inertia of the circle is zero, but nonzero for all
other cases.
Table 4.3
Added-Mass Coefficients for Various Two-Dimensional Bodies.
m11:
πρa2
πρb2
m22:
2
πρa
2
m66:
0
m11:
πρ[a2 + (b2 − a2)2/b2]
m22:
πρa
m66:
*
πρa2
πρa
1
8
2
0
πρ( a − b )
2
2 2
1
8
πρa4
πρa2
4.754 ρa2
2
πρa
4.754 ρa2
ρa4
0.725 ρa4
2
π
*For the finned circle the added moment of inertia is given by the formula
2
2
m66 = ρa4 (π −1 csc 4 α[ 2α 2 − α sin 4α + 21 sin2 2α ] − π /2 ) , where sin α = 2ab/(a + b ) and
π/2 < α < π. The derivations of this result, and of m66 for the crossed fin and square,
are given by Newman (1978). Derivations of the remaining coefficients in this table
are given or cited by Kennard (1967).
The Motion of an Ideal Fluid
153
Table 4.3 lists the added-mass coefficients for several two-dimensional
bodies including the circle, ellipse, flat plate, finned circle, crossed flat
plates, and square. The symmetry of these bodies is such that the remaining added-mass coefficients are zero. Thus in these cases there is no coupling
between the three principal modes of motion of the body. The results shown
in table 4.3 can be derived by appropriate conformal mappings, some of
which are illustrated in the problems listed at the end of this chapter.
In three dimensions, ellipsoids are the most general bodies where comparable analytical results are available. Since there are three planes of symmetry, the only nonzero added-mass coefficients are the six values of mij
where i = j. These will depend on the two nondimensional ratios defining
the shape of the ellipsoid; appropriate graphs may be found in Kochin,
Kibel, and Roze (1964).
A simpler three-dimensional body is the spheroid, or ellipsoid of revolution. If the coordinate x1 is chosen to coincide with the axis of revolution,
with a the semilength of this axis, and b the radius at the equatorial plane
x1 = 0, three nonzero coefficients must be considered, including the longitudinal added mass m11, the lateral added mass m22 = m33, and the rotational
added mass m55 = m66. These will depend on the ratio b/a or the beamlength ratio of the body.
Graphs of the three added-mass coefficients for a spheroid are shown in
figure 4.8. The upper half of this figure shows the coefficients nondimensionalized in terms of the mass and moment of inertia of the displaced volume of fluid. In the limiting case where b/a → 0, or a slender spheroid, the
longitudinal added-mass coefficient tends to zero in a manner analogous to
the flat plate moving in its own plane. On the other hand, the corresponding limits for the lateral added-mass coefficient and added moment of inertia are both equal to 1.0, as one would predict using the strip theory and
the added mass of a circular cylinder. For increasing values of b/a, the longitudinal added mass becomes significant, whereas the lateral added-mass
coefficient decreases because of the three-dimensional effects. The case b/a
= 1 coincides with the sphere, where the translational added-mass equals
one half of the displaced mass, and the rotational coefficients vanish. For
b/a > 1, the spheroid is flattened, or oblate. For b/a → ∞, corresponding to
a circular disc, the coefficients based on displaced mass and mass-moment
are degenerate; to avoid this difficulty the coefficients are renormalized in
the lower half of figure 4.8.
Figure 4.8
Added-mass coefficients for a spheroid, of length 2a and maximum diameter 2b. The
added mass m11 corresponds to longitudinal acceleration, m22 to lateral acceleration
in the equatorial plane, and m55 denotes the added moment of inertia, for rotation
about an axis in the equatorial plane. In the upper figure, the coefficients are nondimensionalized with respect to the mass and moment of inertia of the displaced
volume of the fluid, and in the lower figure with respect to the same quantities for
a sphere of radius b.
The Motion of an Ideal Fluid
155
4.16 The Body-Mass Force
It is useful to note here the similarity between Newton’s equations, for the
force and moment associated with the inertia of the body mass, and the
corresponding hydrodynamic pressure force and moment. This analogy is
of interest for its own sake, and the force and moment due to the body mass
will be required in chapters 6 and 7 to derive the equations of motion for
unrestrained vessels.
If ρB denotes the mass-density of the body, which is a function of position, the inertial force associated with this mass is given by the rate of
change of linear momentum, in the form
F=
d
ρ B (U + W × r ′ ) d .
dt ∫∫∫
B
(135)
Here B denotes integration over the body volume. Similarly, from conservation of angular momentum, the moment is given by
M=
d
ρ Br × (U + W × r ′ ) d .
dt ∫∫∫
B
(136)
This pair of integrals, for the linear and angular momentum of the body,
may be compared with the surface integrals (94–95) which give the corresponding momentum of the fluid.
To take advantage of the subsequent reduction of (94–95), a decomposition of the body velocity can be made, corresponding to the gradient of
(101) for the fluid velocity. For this purpose we define six vectors bj such
that
U + Ω × r′ = U jbj.
(137)
Equating the factors of the six velocity components Uj in (137), it follows
that
( b1 , b2 , b3 ) = ( i , j, k ),
(138)
( b 4 , b5 , b6 ) = ( i , j, k ) × r ′ ,
(139)
where (i, j, k) are the unit vectors parallel to (x, y, z).
The hydrodynamic pressure force and moment are related by (115–116)
to the added-mass coefficients mij, which can be defined in terms of the
156
Chapter 4
kinetic energy of the fluid volume by (119). Thus, by analogy, the coefficients of the body-mass matrix can be defined as
M ij = ∫∫∫ ρ B ( bi ⋅ b j ) d  .
(140)
B
From the definitions (138–139), the matrix of body-inertia coefficients follows in the more conventional form
 m
 0

 0
M ij = 
 0
 mzG

− myG
0
m
0
−mzG
0
mxG
0
0
m
myG
− mxG
0
0
− mzG
myG
I11
I 21
I 31
mzG
0
− mxG
I12
I 22
I 32
− myG 
mxG 

0 
.
I13 
I 23 

I 33 
(141)
Here the body mass is
m = ∫∫∫ ρ B d  ,
(142)
B
the vector position of the center of gravity is
xG =
1
ρB x d ,
m ∫∫∫
B
(143)
and the moments of inertia can be defined by
I ij = ∫∫∫ ρ B[ x′ ⋅ x′δ ij − xi′x′j ] d  ,
(144)
B
where δij is the Kroenecker delta function.
With these definitions and Mij substituted for mij, equations (115–116)
can be used to evaluate the force and moment (135–136). Alternatively, the
total force including the body inertia and fluid pressure can be evaluated
from (115–116), in terms of the virtual-mass coefficients (Mij + mij). Thus
the body behaves in the fluid as it would in a vacuum, but the body-mass
matrix is replaced by the virtual-mass.
4.17 Force on a Body in a Nonuniform Stream
Some of the preceding results can be extended to a body in a nonuniform
stream. The fundamental assumption is that the nonuniformity of the
The Motion of an Ideal Fluid
157
stream is slowly varying, relative to the length scale of the body. A practical
and important example of this situation is the case where the body is acted
upon by an incident wave field, with the restriction that the body is small
compared to the wavelength.
Let us assume that the fluid motion in the absence of the body is
described by a velocity potential Φ(x, t). This may be unsteady in time, but
the length scale characterizing changes in Φ, say Φ / ∇Φ , is assumed large
compared to the dimensions of the body. To treat the simplest case, we
assume that the orientation of the body, with respect to the stream, is such
that the stream velocity at the body is parallel to the x-axis with local velocity U = ∂Φ/∂x, and that the body symmetry is such that with this orientation
of the coordinates, m12 = m13 = 0.
From the kinematic viewpoint, the disturbance of the stream due to the
body depends only on the relative flow between the body and the fluid.
Thus, with the assumptions noted above, the total potential including the
body disturbance is given by
φ = Φ + (U 1 − U )φ1.
(145)
Here U1 is the velocity of the body, and ϕ1 the corresponding velocity potential, as defined by (101). Only the velocity U1 is included here since, with
the symmetry noted above, there is no interaction between the remaining
body velocities and the nonuniformity of the stream.
The pressure force Fx acting on the body in the x-direction may be computed by substituting (145) in (90):
Fx = − ρ
d
[Φ + (U 1 − U )φ1 ]nx dS
dt ∫∫
SB
{
∂Φ
∂φ
∂φ
∂Φ
− ρ ∫∫ 
+ (U 1 − U ) 1  
+ (U 1 − U ) 1 
∂x 
∂n   ∂x
 ∂n
SC
−
(146)
}
1
nx[ ∇Φ + (U 1 − U )∇φ1 ]2 dS.
2
The integral over the body surface SB can be evaluated using Gauss’ theorem for the term involving Φ and (112) for the term involving ϕ1. Since
U = ∂Φ/∂x is effectively constant over the length scale of SB, it follows that
∫∫ [Φ + (U
SB
1
− U )φ1 ]nx dS = − ∫∫∫
B
∂Φ
1
d  + m11(U 1 − U )
ρ
∂x
1
m11U 1 − U ( ∀ + m11 / ρ ).
ρ
(147)
158
Chapter 4
Thus the first term in (146) contributes a force equal to
dU dU 1 
dU
m11 
−
.
+ ρ∀
dt 
dt
 dt
Here dU/dt represents the value of ∂2Φ ( x, t ) / ∂x∂t at the body.
The integral over the control surface SC can be divided into three separate contributions, associated respectively with the stream potential Φ, the
body potential ϕ1, and cross-terms involving both. Since ∇2Φ = 0 throughout the entire region interior to SC, including the interior of SB, (89) can be
applied without SB to show that there is no contribution from the terms
involving Φ alone. Moreover, by the argument leading to (94), there is no
contribution to the control-surface integral from ϕ1 by itself, and only the
cross-terms remain to be considered.
For the contribution from the cross-terms, a straightforward extension
of (89) shows that the control surface SC can be chosen arbitrarily, so long
as it is exterior to the body. We shall locate this surface in the far field of the
body, where the dipole approximation (123) is valid for the potential. If we
use (128) for the dipole moment and consider only the cross-terms, the last
integral in (146) takes the form
∂Φ ∂φ1 ∂Φ ∂φ1
(U 1 − U )∫∫ 
+
− nx∇Φ ⋅ ∇φ1  dS
n
x
x
n
∂
∂
∂
∂


SC
=
1
 ∂Φ ∂ 2  1  ∂Φ ∂ 2  1 
( ∀ + m11 / ρ )(U 1 − U )∫∫ 
+
∂n ∂x 2  r  ∂x ∂n ∂x  r 
4π
SC 
− nx ∇ Φ ⋅ ∇
(148)
∂  1 
dS.
∂x  r  
The right-hand side of (148) can be expressed in the form
1
 ∂2Φ ∂  1 1 ∂ ∂2Φ

( ∀ + m11 / ρ )(U 1 − U )∫∫  2
−
+ (∇ × A ) ⋅ n  dS
4π
∂x ∂n  r  r ∂n ∂x 2

SC 
 ∂2Φ 
= −( ∀ + m11 / ρ )(U 1 − U )  2  ,
 ∂x  r = 0
where the components of the vector A are
Ax = 0,
∂Φ ∂  1  1 ∂2Φ ∂Φ ∂  1 
+
,
−
∂z ∂x  r  r ∂x∂z ∂x ∂z  r 
∂Φ ∂  1  1 ∂2Φ ∂Φ ∂  1 
−
Az =
.
+
∂y ∂x  r  r ∂x∂y ∂x ∂y  r 
Ay = −
(149)
The Motion of an Ideal Fluid
159
Here Laplace’s equation has been used, for Φ and its derivatives as well
as for the source potential; the right-hand side of (149) follows by using
Green’s theorem in the form (76) and noting from Stokes’ theorem that the
integral of (∇ × A ) ⋅ n vanishes over the closed surface SC.
Replacing ∂2Φ/∂x2 at the body by ∂U/∂x, we obtain the total pressure force
on the body in the form
∂U
∂U 
dU 1
Fx = ( ρ∀ + m11 ) 
+ (U − U 1 )
− m11
.
∂x 
dt
 ∂t
(150)
This is the desired expression for the force acting on a body that moves
with velocity U1(t) in the presence of a slowly-varying nonuniform stream
of velocity U(x, t). Significantly, this force is not simply a function of
the relative velocity U − U1. The factor of the added-mass coefficient in
(150) can be written as a function of the relative velocity, but there is
an additional “buoyancy force” proportional to the displaced volume of
the body.
If Euler’s equations (2) are rewritten in a coordinate system translating
with the body velocity U1 the quantity in square brackets in (150) can be
identified with the pressure gradient ∂p/∂x of the nonuniform stream, in the
absence of the body, and (150) can be rewritten in the form
Fx = −( ∀ + m11 / ρ )( ∂p / ∂x) − m11
dU 1
.
dt
(151)
Thus, the force due to the nonuniform stream is equal to the pressure gradient associated with this stream, multiplied by the effective volume ∀ + m11 / ρ .
Note that analogies based on Archimedian hydrostatics would overlook the
contribution from the added-mass coefficient!
From (150) one can derive an equation of motion for the body, by equating Fx to the product of body mass m and acceleration dU1/dt. In particular,
if the body is neutrally buoyant with m = ρ∀, equilibrium of the forces will
occur if U1 = U. Thus the body will be carried along as if it were a fluid
particle.
The more general case where the added-mass tensor mij contains offdiagonal elements was derived originally by G. I. Taylor, using an energy
argument due to Kelvin. This approach is outlined by Lamb (1932).
160
Chapter 4
4.18 The Method of Images
When a body moves in proximity to a plane rigid boundary, such as a horizontal or vertical wall, the interaction between the two must be accounted
for. Formally, this is done by imposing the boundary condition ∂ϕ/∂n = 0 on
the wall and attempting to solve the resulting boundary-value problem for
the prescribed motion of the body. A convenient approach is to replace the
wall by an image body, symmetrically disposed on the opposite side of the
wall with a suitably prescribed symmetric motion to ensure that the boundary condition on the wall is satisfied. In this sense body-wall interactions
are closely related to the interactions between two adjacent and identical
bodies. For example, the interaction between a ship and an adjacent vertical canal bank is identical to the interaction of the same ship hull with a
catamaran twin-hull.
If a two-dimensional source is situated at a point (0, b) above the x-axis
and an image source is situated at (0, −b), the resulting complex source
potential is simply
F=
m
log( z 2 + b2 ).
2π
(152)
Differentiation gives a dipole plus its image, in the x-direction,
F=
µ  1
1 
,
+
2π  z − ib z + ib 
(153)
but the corresponding vertical dipole is obtained by differentiation with
respect to the source point b as opposed to the vertical coordinate y.
From (153), the flow past a pair of circular cylinders of radius a can be
approximated by
F = Uz +
Ua2
Ua2
.
+
z − ib z + ib
(154)
This solution is not exact, because the flow associated with the image dipole
will distort the streamlines moving past the first dipole, but this effect will
be small if the ratio a/b is small. The corresponding result for a pair of
spheres, or for a single sphere moving near a wall, is readily obtained.
This procedure can be extended to a body or source situated between
two parallel walls, say at y = ± 21 b. In this case, to satisfy the boundary
The Motion of an Ideal Fluid
161
conditions on both walls simultaneously, it is necessary to use an infinite
row of images, at y = ± b, ± 2b, ± 3b, … . In two-dimensional problems, the
resulting infinite series for sources and dipoles can be summed in closed
form, or the solutions can be guessed by noting that these must be periodic functions of y with period b. Thus, the complex source potential is
given by
F=
πz
m
log sinh
,
2π
b
(155)
and the flow past a (distorted) circle is given by the superposition of a free
stream and dipole, in the form
F = Uz + πU ( a2 / b )coth π z / b.
(156)
These problems are discussed by Lamb (1932), who notes that the distortion of the circle is negligible provided its diameter is less than half of the
space between the walls.
The problem of a vertical flat plate, situated midway between two parallel horizontal walls, can be solved in closed form, as outlined by Sedov
(1965). The added-mass coefficient in this case is given by the expression
m11 = −2 ρ
πa
b2
log cos
,
π
b
(157)
where 2a is the breadth of the plate. This result is exact, and the two limiting cases are a single flat plate in an infinite fluid when a/b → 0 and a completely blocked channel with infinite added mass when a/b → 1/2. Similar
results for rectangular profiles are given in numerica1 form by Flagg and
Newman (1971).
Wall corrections for wind tunnels and water tunnels can be derived in
this manner. For three-dimensional motion in a rectangular tunnel, a doubly infinite array of image bodies or singularities is required.
Problems
1. Show that the velocity potentials defined by the real parts of the complex
potentials in figures 4.4 and 4.5 satisfy the correct boundary conditions on
the boundaries of each flow. Assuming each of these flows is steady, what
is the corresponding expression for the pressure throughout the fluid and
along the boundaries?
162
Chapter 4
2. Show that the maximum tangential velocity for the streaming flow past
a circle is twice the free-stream value, and for a sphere is one and one half
the free-stream value. What is the corresponding result for a Rankine ovoid
at the equatorial plane? How does this compare with the sphere as the distance between the source and sink goes to zero? What is the opposite limit
as the separation distance becomes large?
3. For a Rankine half-body generated by the combination of the uniform
stream ϕ = Ux and a three-dimensional source at the origin, find the equation of the dividing streamline in terms of the stream function. Show that
the distance from the source to the stagnation point is one half of the limiting radius far downstream and that the body’s radius in the plane x = 0 is
the geometric mean of these two lengths. Show that the tangential velocity
at x = 0 exceeds the free-stream velocity by a factor (5/4)1/2. Sketch the tangential velocity as a function of position along the body surface.
4. A circular piston of radius a moving with axial velocity U is mounted
flush in the plane of a rigid baffle extending to infinity in all radial directions in the plane x = 0. The fluid occupies the region x > 0. State the
complete boundary-value problem for this situation. Using Green’s theorem, give an appropriate source distribution representing this flow. What
is the limiting form of the velocity potential at large distances from the
piston? (Assume that the departure of the piston face from the plane x = 0
is negligible.)
5. Use the complex potential (57) to derive the added-mass coefficient of
a flat plate, using (114). Show that the same result can be obtained from
(130).
6. Show that the analytic function
F ′( z ) = u − iv = −( 21 a2 − z 2 )( a2 − z 2 )−1/2 − iz
satisfies the correct boundary condition for the rolling flat plate shown in
table 4.3 and behaves correctly at infinity. Using equation (114) and integrating by parts, verify that
a
a
−a
−a
m66 = −2 ρ ∫ xφ( x, 0 + ) dx = − ρ ∫ ( a2 − x2 ) F ′( x ) dx =
π 4
ρa .
8
7. Two common approximations for the added-mass coefficient m11 or m22
of two-dimensional bodies are (1) the displaced mass and (2) the displaced
The Motion of an Ideal Fluid
163
mass of a circle with the same projected width normal to the direction
of acceleration. Compare these for the bodies shown in table 4.3. Which
approximation is, in general, the most accurate?
8. A naive ocean engineer, infatuated with strip theory and ignorant of
three-dimensional effects, is required to compute the added-mass coefficients for a circular disc of radius b and negligible thickness. Using the correct result shown in figure 4.8, show that m11 will be overestimated by a
factor of about 1.6, and the error for the added moment of inertia depends
on whether the “strips” are parallel or normal to the axis of rotation.
9. By resolving the translation of a flat plate into components normal and
tangential to its plane, show that if it is inclined at an angle θ from the
x-axis, the added-mass coefficients are
m11 = πρa2 sin2 θ
m22 = πρa2 cos2 θ
π
m12 = m21 = ρa2 sin 2θ
2
What is the added moment of inertia?
10. Show that the added mass coefficients m11 and m22 in table 4.3 for a
crossed fin and square are unchanged, if the body is rotated through an
arbitrary angle, and that m12 = m21 = 0.
11. Show that the conformal transformation (51), with U = 1, maps the
circle of radius r0 onto an ellipse in the ζ-plane, with semi-axes r0 + A/r0 and
r0 − A/r0. Using the complex potential for the flow past a circle, determine
the effective dipole moments for the ellipse, and use (130) to derive the
added-mass coefficients m11 and m22.
12. From qualitative estimates of the relative magnitudes of the added
mass coefficients and from (116), show that a streamlined body is generally
unstable when moving with constant velocity parallel to its long axis. Use
the same argument to show that leaves falling from trees will generally be
horizontal.
13. The velocity potentials for internal motions within the interior of a
closed surface S, enclosing a volume of fluid, can be defined as in (101–103).
Show that the solutions of the three internal translational potentials are
given simply by ϕi = xi, whereas the solutions for rotational motions are
164
Chapter 4
more complicated. By writing Green’s theorem for the region exterior to
the surface S, show that if ϕI and ϕE are solutions of Laplace’s equation in
the interior and exterior domains, respectively, then the exterior potential
can be expressed as a distribution of sources on S, of strength (∂/∂n) (ϕI −
ϕE), provided ϕI satisfies the boundary condition ϕI = ϕE on S. Show that the
exterior potential can also be represented as a dipole distribution normal to
S, of moment ϕI − ϕE, provided ∂ϕI/∂n = ∂ϕE/∂n on S. Use the latter representation to derive (128).
14. Compare the relative magnitudes of the forces acting on a stationary
prolate spheroid, with length-diameter ratio equal to 5.0, in the presence
of an accelerating stream that is (1) parallel and (2) perpendicular to the
longitudinal axis of the spheroid.
15. Using equation (90), derive an integral expression for the attraction
force that occurs when a body moves with steady translation parallel to an
infinite plane rigid wall, in terms of the pressure force on the wall. Show
that this force always acts on the body in the direction toward the wall.
If the body is sufficiently far from the wall so that the potential can be
approximated by the dipole (123) plus an image dipole on the opposite side
of the wall, show that the attraction force is given by
F=
3π ρU 2 A112
3 ρU 2
( ∀ + m11 / ρ )2 ,
=
d4
4
64π d 4
provided A12 = A13 = 0. Here d is the distance from the body axis to the wall.
Derive from this equation and (133) the special case
F=
3π
ρU 2 R6 / d 4
16
for a sphere of radius R, and
F=
3
ρU 2 ∀2 / d 4
64π
for a slender prolate spheroid moving in the longitudinal direction.
References
Cummins, W. E. 1957. The force and moment on a body in a time-varying potential
flow. J. Ship Res. 1 (1): 7–18.
The Motion of an Ideal Fluid
165
Flagg, C., and J. N. Newman. 1971. Sway added-mass coefficients for rectangular
profiles in shallow water. J. Ship Res. 15:257–265.
Havelock, T. H. 1963. Collected papers. ONR/ACR-103. Washington: U. S. Government Printing Office.
Hildebrand, F. B. 1976. Advanced calculus for applications. 2nd ed. Englewood Cliffs,
N.J.: Prentice-Hall.
Kennard, E. H. 1967. Irrotational flow of frictionless fluids, mostly of invariable density.
David Taylor Model Basin Report 2229.
Kochin, N. E., I. A. Kibel, and N. V. Roze. 1964. Theoretical Hydromechanics. English
translation of. 5th Russian ed. New York: Wiley.
Lamb, H. 1932. Hydrodynamics. 6th ed. Cambridge: Cambridge University Press.
Reprinted 1945, New York: Dover.
Landweber, L., and C. S. Yih. 1956. Forces, moments, and added masses for Rankine
bodies. Journal of Fluid Mechanics 1:319–336.
Milne-Thomson, L. M. 1968. Theoretical hydrodynamics. 5th ed. New York:
Macmillan.
Morse, P. M., and H. Feshbach. 1953. Methods of theoretical physics, 2 vols. New York:
McGraw-Hill.
Newman, J. N. 1978. The added moment of inertia of two-dimensional cylinders. J.
Ship Res. 23:1–8.
Sedov, L. I. 1965. Two dimensional problems in hydrodynamics and aerodynamics.
English translation. New York: Interscience.
5 Lifting Surfaces
Chapter
Lifting
5
Surfaces
© Massachusetts Institute of TechnologyAll Rights Reserved
One of the most striking applications of fluid mechanics is the lifting surface that supports the flight of birds, airplanes, and hydrofoil boats. Similar
effects are involved in the actions of control surfaces such as rudders, yacht
sails and keels, and screw propellers. Thus, there are many applications of
lifting surfaces in marine hydrodynamics; a proper understanding of their
actions must preface an intelligent approach to their design and use.
Typically, a lifting surface is a thin streamlined body that moves through
the surrounding fluid at a small angle of attack, with a resultant hydrodynamic lift force generated in the direction normal to the forward movement. We shall suppose that the lifting surface is primarily oriented in the
x-z plane and moving with constant velocity U in the positive x-direction,
as shown in figure 5.1. The lift force L is the component of the hydrodynamic force parallel to the y-axis; the drag force D acts in the negative
x-direction.
The principal geometric dimensions of the lifting surface are its chord
length l and span s, measured parallel to the x- and z-axes respectively. For
a rectangular planform, l is a constant, but in general the chord l(z) will
depend on the transverse coordinate. The ratio of the span to the mean
chord is the aspect ratio A, given by
A = s2 / S ,
(1)
where S is the projected area of the lifting surface on the plane y = 0.
The aspect ratio is an important measure of the effect of threedimensionality. A large aspect ratio implies that the flow will be nearly independent of the transverse coordinate z, and a two-dimensional strip-theory
approach is valid. This situation is exemplified by older aircraft wings, as
well as by some hydrofoil configurations and deep narrow centerboards on
168
Chapter 5
Figure 5.1
Three-dimensional lifting surface.
small sailing craft. On the other hand, if the aspect ratio is smaller, as in the
case of most rudders and control surfaces, three-dimensional flow effects
will be important and must be included in our analysis.
Lifting surfaces of the type described are planar. Excluded from this category is the screw propeller illustrated in figure 2.8. The results to be derived
for planar foils are qualitatively applicable to the blades of a propeller. In
particular, the two-dimensional theory can be used in a strip-theory sense
to describe the flow past each section of a propeller blade, and the threedimensional flow past a propeller will include trailing vortex sheets similar
to those existing in the planar case. The principal distinctions are that the
propeller blades and associated vortex sheets are roughly helical, and the
blades affect each other in a manner analogous to a cascade of planar foils
operating in proximity. Detailed descriptions of propeller theory are given
by Cox and Morgan (1972) and Kerwin (1973).
The case of two-dimensional flow past a lifting surface or hydrofoil will
be considered first to provide a quantitative understanding of lifting surfaces of high aspect ratio, as well as qualitative understanding of the more
general case. Subsequently, we shall consider the more complicated threedimensional case of finite aspect ratio planar lifting surfaces. Throughout
our discussion we will assume that viscous effects are confined to a thin
boundary layer along the surface of the hydrofoil and that the foil surface
is itself thin and situated at a small angle of attack relative to the incident
Lifting Surfaces
169
flow. Thus, in the jargon of aerodynamics, we shall restrict our discussion
to the case of linearized thin-wing theory.
5.1 Two-Dimensional Hydrofoil Theory
Let us consider the steady flow of fluid past a thin streamlined section
asymmetrical about y = 0. This asymmetry may be due either to a small
angle of attack a between the axis of the hydrofoil and the x-axis, or to the
existence of curvature or camber of the hydrofoil, or to a combination of
these two effects. Under these circumstances the streamlines will appear
roughly as shown in figure 5.2. In particular, viscous effects are confined to
a thin boundary layer adjacent to the surface of the foil, and the flow will
leave the trailing edge in a smooth tangential manner.
For viscous effects to be confined to a thin boundary layer, not only
must the Reynolds number be reasonably large but also the body must be
sufficiently thin and streamlined that separation does not occur. If the body
is not thin or if the angle of attack is too large, the flow will be typified by
the sketches shown in figure 5.3. Thus, there are practical limits to both the
body shape and the angle of attack.
The additional assumption that the flow leaves the trailing edge in a
smooth tangential manner (as shown in figure 5.2) is fundamental to the
theory of lifting surfaces. Another possibility that can be envisaged is shown
in figure 5.4; this involves an infinite velocity around the trailing edge and
a stagnation point on the upper (or possibly the lower) surface of the foil.
In fact, the streamlines as drawn in figure 5.4 correspond to a potential flow
without circulation, whereas the expected situation shown in figure 5.2
Figure 5.2
Assumed flow past foil.
170
Chapter 5
Figure 5.3
Separated flow due to a bluff body (left) or excessive angle of attack (right).
Figure 5.4
Flow past foil without circulation.
is obtained from that in figure 5.4 by adding a positive counterclockwise
circulation around the foil, sufficient to move the stagnation point back to
the trailing edge. This may seem to contradict Kelvin’s theorem that the
circulation is constant about any material contour always containing the
same fluid particles, since this constant must equal zero if the fluid motion
started from a state of rest.
This apparent contradiction can be resolved by noting that a material
contour surrounding the foil cannot be related to another that is initially
upstream, since the foil will “pierce” the fluid on the upstream contour.
The contour that is initially upstream can only be related to a subsequent
contour that surrounds the foil and a portion of its wake, as shown in figure
5.5, and the circulation about this contour will indeed remain zero. A more
illuminating contour is one sufficiently large to surround the foil initially
and at all subsequent times, as shown in figure 5.6. Here the circulation
must be zero, and the circulation about the foil must, therefore, be canceled
by an equal and opposite starting vortex shed from the trailing edge into the
wake during the initial acceleration. The shedding of this starting vortex
can be observed by accelerating a flat plate in a kitchen sink. If the plate is
Lifting Surfaces
171
Figure 5.5
The material contour shown here is initially upstream of the foil, at time t = t1. A
short time later (t = t2), it has surrounded the foil and a portion of the wake (- - - - -).
From Kelvin’s theorem, and the initial conditions upstream, the circulation around
this contour is zero for all times.
Figure 5.6
The material contour shown here is sufficiently large to surround the foil at its original position of rest, and subsequently after acceleration to a steady velocity U. The
circulation about this contour is zero, so that the foil must shed a starting vortex of
strength Γ, equal and opposite to the circulation of the foil.
at right angles to the flow, vortices will be shed from both edges, but if it is
at a small angle to the flow, only the trailing edge will shed a vortex.
If our interest is in the steady-state hydrofoil problem, the starting vortex can be disregarded, since it will be situated infinitely far downstream
and, in reality, will be dissipated by viscous diffusion. Thus, for the twodimensional steady-state problem, it is reasonable to expect irrotational
flow throughout the surrounding fluid, but with a net circulation about
the foil. This circulation is essential to the development of a lift force since
172
Chapter 5
without it there can be no force acting on a body in a steady-state translation. In this respect, the smooth local flow at the trailing edge is vital to the
development of the desired lift force.
The assumption of smooth tangential flow at the trailing edge is imposed
mathematically by the Kutta condition requiring the velocity at the trailing
edge to be finite. This extra condition is added to the conventional statement of the boundary-value problem, which may now be formulated. The
conditions on the fluid flow are that the velocity vector should be equal to
the free-stream velocity −Ui at infinity, tangential to the surface of the foil,
and finite at the trailing edge.
It is convenient to subtract the free-stream component and utilize the
perturbation velocity components (u, v) with the understanding that the
total velocity vector of the two-dimensional flow is (u − U, v). If the perturbation velocity potential ϕ(x, y) is defined such that
(2)
( u , v ) = ∇φ ,
the following boundary-value problem results:
∇ 2φ = 0 ,
throughout the fluid ,
(3)
∂φ / ∂n = Unx , on the foil,
(4)
∇φ < ∞ ,
at the trailing edge,
(5)
∇φ → 0,
at infinity.
(6)
5.2 Linearized Two-Dimensional Theory
The boundary condition (4) on the foil can be simplified under the assumption that the foil is thin and nearly horizontal. Thus we now assume that
the vertical coordinates of the upper and lower foil surfaces, y = yu and
y = yl, respectively, are both much smaller than the chord length l and that
the slopes yu′ ( x) and yl′( x) are small compared to one. For convenience, we
will take the origin of the coordinate system so that the leading and trailing
1
2
edges are situated at x = ± l , respectively, as shown in figure 5.7.
The boundary condition (4) on the foil can be expressed in a more convenient form by utilizing the substantial derivative (3.19). Thus, following
the fluid particles on the upper surface of the foil, where y − yu(x) = 0,
Lifting Surfaces
173
Figure 5.7
Notation for two-dimensional foil.
D
[ y − y u ( x)] = V ⋅ ∇[ y − y u ( x)],
Dt
(7)
∂φ
∂φ
= y u′ ( x)  − U  , on y = y u ( x).
∂y
 ∂x

(8)
0=
or
A similar boundary condition applies on the lower surface y = yl.
The boundary condition (8) shows that the vertical velocity component,
and hence the perturbation potential ϕ, will be roughly proportional to the
slope of the foil. For sufficiently small values of yu′ , the magnitude of the
free-stream velocity −U will be much larger than the horizontal component
of the perturbation velocity; in that case (8) can be approximated by
∂φ / ∂y −Uy u′ ( x), on y = y u ( x),
(9)
which is valid provided |∂ϕ/∂x| ≪ U. Equation (9) is a first-order approximation to (8), in the sense that the perturbation velocity is proportional to
the slope yu′ , and the neglected term yu′ ( ∂φ / ∂x) will be of second order in
the slope.
The boundary condition on the foil can be further simplified, consistent
with the first-order approximation (9). Since the vertical coordinates of the
foil surface differ from zero by a small quantity, the exact surface of the
foil may be collapsed onto the cut (−l/2, l/2) of the real axis, with the upper
174
Chapter 5
surface of the foil corresponding to the upper surface y = 0+ of the cut, and
conversely for the lower surface. Since singularities may exist on or inside
the exact foil, these will be transferred to the cut in a corresponding manner. Then, provided ∂ϕ/∂y is regular outside the foil, the difference between
its value on the exact surface y = yu and the upper side of the cut y = 0+ will
be a second-order quantity, i.e., from a Taylor series expansion,
( ∂φ / ∂y )y = yu = ( ∂φ / ∂y )y = 0 + + y u ( ∂2φ / ∂y 2 )y = 0 + + ... ( ∂φ / ∂y )y = 0 + .
(10)
Combining the approximations (9) and (10) gives the first-order linearized boundary condition on the upper surface of the foil, in the form
∂φ
= −Uy u′ ( x), on y = 0 + , − l 2 < x < l 2 .
∂y
(11)
An identical argument gives the corresponding boundary condition
∂φ
= −Uyl′( x), on y = 0 −, − l 2 < x < l 2 ,
∂y
(12)
on the lower surface. These linearized boundary conditions effectively
replace the foil by a suitable distribution of the vertical velocity on the corresponding cut of the x-axis.
Before attempting to solve this linearized boundary-value problem, let
us consider the corresponding first-order approximation for the hydrodynamic pressure force and moment. From the steady form of Bernoulli’s
equation (4.26), the dynamic pressure in the fluid is given by
1
1
p − p∞ = − ρ( V ⋅ V − U 2 ) = − ρ( u2 + v 2 − 2uU ).
2
2
(13)
Neglecting the nonlinear terms of second-order in the perturbation velocity gives
p − p∞ ρuU .
(14)
Thus, the linearized pressure is simply proportional to the horizontal perturbation velocity component u = ∂ϕ/∂x.
The vertical lift force L acting on the foil is obtained, to the same order
of approximation, by integrating the jump in pressure across the cut. Thus,
it follows that
L=
∫ ( p − p∞ ) dx = ρU ∫ u dx = ρU Γ ,
(15)
Lifting Surfaces
175
where Γ is the total circulation around the foil as defined by (4.4). Equation (15) is a derivation, for linearized thin foils, of the more general KuttaJoukowski theorem. This theorem states that for any two-dimensional body,
moving with constant velocity in an unbounded inviscid fluid, the hydrodynamic pressure force is directed normal to the velocity vector and is equal
to the product of the fluid density, body velocity, and the circulation about
the body (see problem 3).
The hydrodynamic moment M about the z-axis is given by the corresponding integral
M = ρU ∫ ux dx.
(16)
Now, to solve the boundary-value problem for the thin two-dimensional
hydrofoil, it is convenient to decompose the velocity potential into even
and odd functions of y:
φ( x, y ) = φe ( x, y ) + φo ( x, y ),
(17)
1
φe ( x, y ) = φe ( x, − y ) = [φ( x, y ) + φ( x, − y )],
2
(18)
1
φo ( x, y ) = −φo ( x, − y ) = [φ( x, y ) − φ( x, − y )].
2
(19)
The boundary conditions on y = 0± take the form
1
∂φ e
= ∓ U ( y u′ − yl′), on y = 0 ± ,
∂y
2
(20)
∂φ o
1
= − U ( y u′ + yl′), on y = 0 ± .
∂y
2
(21)
Since the operator ∂/∂y is odd, ∂ϕe/∂y is odd and ∂ϕo/∂y is even with respect
to y.
The even and odd potentials correspond to two distinctly different physical problems. From (20), ϕe is the potential for a symmetrical strut, of thickness yu − yl, at zero angle of attack. On the other hand, from (21), ϕo is the
potential of an asymmetric flow, past an arc of zero thickness defined by the
1
2
curve y = ( yu + yt ), or the mean-camber line. Both problems are illustrated
in figure 5.8.
The original problem has now been decomposed into two parts, one representing the effects of thickness and the other representing the effects of
176
Chapter 5
Figure 5.8
Definitions of thickness and lifting problems.
camber and angle of attack. Because the pressure distribution is symmetric
in the thickness problem, there can be no lift force or moment in this case;
hence thickness has no direct effect on the lift and moment. Thickness has
practical effects only when modifications of the pressure distribution affect
separation or cavitation. In particular, the effect of thickness will be important near the leading edge, where the mean-camber line contains a sharp
edge, not present in the real foil with finite leading-edge radius. Thus, in
the flow past the mean-camber line we must anticipate an infinite velocity
at the leading edge which will not occur in practice because of thickness
effects, but which, if it did occur, would be devastating from the standpoints of separation and cavitation. It is therefore permissible, in the lifting
problem of the mean-camber line, to ignore an infinite velocity at the leading edge, on the premise that this singularity will be removed by the effects
of thickness. An entirely different situation will occur at the sharp trailing
edge; an infinite velocity here is prevented by viscous effects in a real fluid
and by the Kutta condition in the ideal-fluid problem.
Since the thickness and lifting problems can be separated, and since the
lift force and moment are independent of the thickness, we will examine
primarily the solution of the lifting problem. The corresponding solution
of the thickness problem, in terms of a source distribution, will be described
briefly in section 5.6.
5.3 The Lifting Problem
Here we focus our attention on the flow past the mean-camber line and
the resulting lift force and moment. It is convenient to denote the vertical
1
2
coordinate of the mean-camber line by η = ( yu + yl ). The corresponding
boundary condition on the cut is
Lifting Surfaces
177
Figure 5.9
Tangential velocity induced by a point vortex around a circle of radius r.
v=
∂φ
= −U η ′( x).
∂y
(22)
From (22), the vertical velocity across the cut is continuous, but a discontinuity in the horizontal velocity component u must be anticipated. This
is emphasized by the linearized form of Bernoulli’s equation (14) and the
Kutta-Joukowski theorem (15), where the presence of a lift force requires a
discontinuity in the horizontal velocity component and a nonzero circulation about the foil section. These considerations suggest a distribution
of vortices along the cut, for such a distribution will possess a nonzero
circulation so long as the total integrated strength of the vortices is not
zero. Moreover, a vortex distribution can be constructed to give any desired
discontinuity of the horizontal velocity on the cut.
The velocity potential of a point vortex, of circulation γ situated at the
point x = ξ on the x-axis, is
φ = Im
γ
γ
 y 
log( x − ξ + iy ) =
tan −1 
.
 x − ξ 
2π
2π
(23)
The corresponding velocity components are
u=
∂φ
γ
−y
=
,
∂x 2π ( x − ξ )2 + y 2
(24)
v=
∂φ
γ
(x − ξ)
=
.
∂y 2π ( x − ξ )2 + y 2
(25)
Thus, the vortex induces a tangential velocity, vt = γ/2πr, where r is the
radial distance from the vortex, as shown in figure 5.9.
178
Chapter 5
If vortices are distributed along the x-axis between the leading and trailing edges of the foil with local circulation density γ(ξ), the resulting velocity
components are given by the integrals
u( x, y ) = −
v ( x, y ) =
1
2π
l2
γ (ξ )y d ξ
,
( x − ξ )2 + y 2
(26)
γ (ξ )( x − ξ ) dξ
.
( x − ξ )2 + y 2
−l 2
(27)
1
2π
∫
−l 2
l2
∫
To show that this vortex distribution can be used to solve the linearized
lifting problem, it is necessary to examine the limiting values of the velocity components defined by (26) and (27) as the point (x, y) approaches the
foil surface. These two limits must be considered separately.
First, let us consider the limiting value of the horizontal velocity, as
defined by (26), when y = ±ε, where ε is positive and ε ≪ l. Then
u( x, ± ε ) = ∓
1
2π
l2
∫
−l 2
γ (ξ )ε
dξ.
( x − ξ )2 + ε 2
(28)
As ε → 0 the integrand tends to zero, except for the point ξ = x, where
the integrand tends to infinity. In this limit, the factor multiplying γ
behaves like a Dirac delta function, extracting the value of γ(x) at the point
ξ = x. More specifically, since the integrand vanishes for ξ ≠ x, (28) can be
replaced by
l2
u( x, ± ε ) ∓
1
ε
γ ( x) ∫
dξ
2π
( x − ξ )2 + ε 2
−l 2
=∓
1
ξ − x 
γ ( x) tan −1 
 ε   −l 2
2π

l2
(29)
1
∓ γ ( x),
2
where the approximations are valid for sufficiently small values of ε/l. Thus,
the limiting values of the horizontal velocity component on the upper and
lower surfaces of the foil are given by the last line of (29), and there will
exist a discontinuity in this velocity component precisely equal to the local
vortex strength γ(x). This confirms our hypothesis that the lifting problem
for the mean-camber line can be represented by a vortex distribution. To
relate the local vortex strength γ to the geometry of the foil, we must invoke
Lifting Surfaces
179
the boundary condition (22) on the surface of the foil; hence we must consider the limiting form of the vertical velocity component.
From (27) the vertical velocity component on the cut is given by the
integral
l2
v ( x, 0 ± ) = −
γ (ξ )
1
−
dξ ,
2π −∫l 2 ξ − x
−
1
1
l < x < l,
2
2
(30)
where the integral is to be interpreted in the sense of the Cauchy principal
value, or
b
f (ξ )
b
 x − ε f (ξ )
f (ξ ) 
dξ + ∫
dξ  .
→0
ξ
ξ
−
− x 
x
a
x+ε
 ∫−
∫− ξ − x dξ ≡ lim
ε

a
(31)
(The legitimacy of this interpretation is confirmed in section 5.6.)
If the geometry of the mean-camber line η(x) is prescribed, the vertical
velocity v(x, 0) can be computed from the boundary condition (22). In this
case, (30) is an integral equation since the unknown function γ appears in
the integrand. We will see in section 5.7 that the general solution of this
integral equation is given by the expression
γ ( x) =
2
π[(l 2 )2 − x2 ]1 2
 l 2 v(ξ , 0 )[(l / 2 )2 − ξ 2 ]1 2

dξ + 1 2 Γ  ,
 ∫−
−
x
ξ
 − l 2

(32)
where the total circulation about the foil is
l2
Γ=
∫
γ (ξ ) dξ.
(33)
−l 2
At this stage Γ is unknown and the solution (32) is not unique, even
though it has been chosen to satisfy the boundary condition (22) on the
foil. This is not surprising, since we have not yet imposed the Kutta condition at the trailing edge. The general case with arbitrary circulation will
include an infinite velocity at the trailing edge, as in the case shown in
figure 5.4, differing from the correct solution in figure 5.2 by the appropriate circulation.
In general, the vortex strength γ, and hence the horizontal velocity component on the foil, will be infinite both at the leading and trailing edges as
a result of the square-root infinities in (32). This can be avoided at the trailing edge only if Γ is chosen so that the quantity in braces in (32) vanishes
180
Chapter 5
at x = −½l . Imposing the Kutta condition results in the following integral for
the circulation:
12
l2
Γ = −2
 ½l − ξ 
v (ξ , 0 ) 

 ½l + ξ 
−l 2
∫
(34)
dξ.
This is a particularly useful result because the lift is given by (15), in terms
of the circulation, and substituting (34) in (15) gives an equation for the
lift as
12
l2
L = 2 ρU 2
dη  ½l − ξ 
dξ  ½l + ξ 
−l 2
∫
(35)
dξ.
Here the boundary condition (22) has been utilized to substitute the slope
of the mean camber line for the vertical velocity component. Alternatively,
the nondimensional lift coefficient is
12
l2
CL =
L
4
dη  ½l − ξ 
= −
½ρU 2l l −∫l 2 dξ  ½l + ξ 
dξ.
(36)
The moment acting on the foil can be determined similarly from (16),
(29), and (32). Neglecting the details of the integration, we get as the final
result
l2
M = ρU
∫
γ ( x ) x dx = 2 ρU 2
−l 2
l2
12
dη
(½l )2 − ξ 2  dξ ,
d
ξ
−l 2
∫
(37)
and the nondimensional moment coefficient corresponding to (36) is
CM =
M
4
=
½ρU 2l 2 l 2
l2
12
dη
(½l )2 − ξ 2  dξ.
ξ
d
−l 2
∫
(38)
The center of pressure xCP is the ratio xCP = M/L.
5.4 Simple Foil Shapes
To illustrate these results, we consider first the simplest case of a flat plate,
or uncambered foil, with an angle of attack dη/dx = α. Then, it follows from
(36) that
CL = 2πα ,
and from (38) that
(39)
Lifting Surfaces
181
CM = ½πα ,
(40)
Thus, the lift coefficient of an uncambered foil is 2π times the angle of
attack, and the center of pressure is at the quarter-chord point, xCP = l/4. As
an indication of the accuracy of the linearized theory, the exact lift coefficient of a flat plate in the absence of separation is
CL = 2π sin α
(41)
(see problem 5). Thus, the error associated with the linearized theory is
insignificant at realistic angles of attack. Similar conclusions follow from
the experimental data shown in figure 2.6, where the curve CL(α) is virtually linear for angles of attack less than 12–15 degrees. For larger angles
separation or stall occurs, with much greater significance than any errors
resulting from the linearization assumption.
The vortex distribution corresponding to the flat plate can be obtained
by integrating (32) with v = −αU. The result is
12
½l + x 
γ ( x ) = 2αU 
 ½l − x 
,
(42)
where (81) and (84) have been used. The function (42) is displayed in
figure 5.10.
Figure 5.10
The vortex distribution for a fiat plate, given by equation (42).
182
Chapter 5
Since the linearized pressure jump across the foil is proportional to the
local vortex strength, figure 5.10 can be regarded as a graph of the pressure
loading on the foil. This loading is concentrated near the leading edge and,
in fact, becomes infinite as x → ½l .
The severe pressure gradient near the leading edge will cause separation
or cavitation as the angle of attack is increased. Nevertheless, the flat-plate
mean-camber line represents the lifting problem for all symmetric foils
since the thickness distribution does not affect the lift and moment. Thus,
these results apply to several cases of practical importance, notably for rudders, keels, and stabilizing fins, where equal performance is required for
positive and negative angles of attack.
When the foil has a preferred direction of lift, the mean-camber line can
be modified to make the loading more uniform. The simplest possibility is
a parabolic-arc mean-camber line, with slope
(43)
dη dx = α − β x.
Here α and β are constants; α is the angle of attack of the incoming stream
relative to the straight line connecting the leading and trailing edges, and
βl2/8 is the elevation of the foil at mid chord above this straight line.
Since equations (32, 35, 37) for the vortex strength, lift, and moment are
linear functions of the slope dη/dx, the two terms in (43) can be integrated
separately and the results superposed. The angle-of-attack term will give
results identical to the flat plate, whereas the effects of the camber will be
independent of the angle of attack. This decomposition of the camber and
angle of attack applies not only for the parabolic arc, but for any cambered
foil. In particular, the lift coefficient CL(α) can be written in general in the
form
(44)
CL (α ) = CL ( 0 ) + 2πα ,
where the term CL(0) represents the effects of camber, and the term 2πα
represents the effect of the angle of attack. A similar result applies for the
moment.
Proceeding with our analysis of camber for a parabolic arc, we can set
the angle of attack equal to zero in (43), in view of the decomposition (44).
Evaluating the integrals (32), (35), and (37), we get
12
γ ( x ) = 2U β (½l )2 − x2 
,
(45)
Lifting Surfaces
183
CL = ½πβl ,
(46)
CM = 0.
(47)
For nonzero angles of attack, the corresponding results (42), (39), and (40)
can be superposed.
The vortex strength (45) is indeed more uniform than that of the flat
plate, and the leading-edge singularity has been removed. This feature is
common to all mean-camber lines symmetric about the mid-chord point,
since the resulting vortex strength must be symmetric in x and thus will
vanish not only at the trailing edge but also at the leading edge. By the
same argument, the center of pressure must be at the mid-chord point for
such foils, and (47) confirms this for the parabolic arc.
Since a nonzero angle of attack will add the flat-plate load distribution to
the cambered foil, the leading-edge singularity will return in this case, and
only at the ideal angle of attack will there be no leading-edge singularity.
The ideal angle of attack corresponds to the situation where the forward
stagnation point of the foil coincides with the leading edge, and the slopes
of the dividing streamline and the mean-camber line are equal at this point.
For mean-camber lines symmetrical about the mid-chord point, the ideal
angle of attack is zero, but for more general foil shapes, the ideal angle of
attack can be nonzero.
The parabolic arc represents a substantial improvement over the flat
plate from the standpoint of uniform loading and pressure distribution, but
further improvement is possible if more complicated mean-camber lines
are considered. Rather than proceed to analyze alternative foil shapes, however, it is more expedient to approach the optimum loading distribution
problem by finding the shape of the mean-camber line such that the loading is a constant. This indirect or design problem, with a specified load distribution, is actually simpler to solve, since (30) becomes a definite integral
instead of an integral equation with an unknown integrand. In the present
case, with γ = constant, and after invoking the boundary condition (22) for
the vertical velocity, we have
l2
dξ
½l − x
∂η
γ
γ
−
=
=
log
.
½l + x
∂x 2πU −∫l 2 ξ − x 2πU
(48)
184
Chapter 5
Integrating with respect to x, we obtain the desired mean camber
η=−
γ
{(½l + x) log (½ + x l ) + (½l − x) log (½ − x l )}.
2πU
(49)
From (48) it is apparent that requiring uniform load distribution has led
to a mean-camber line with infinite slope at both ends. This is not a serious
objection at the leading edge where the effects of thickness will alleviate the
singularity, but the infinite slope at the trailing edge causes flow separation
or cavitation and hence should be avoided. This difficulty might have been
anticipated, since the Kutta condition cannot be satisfied unless the loading vanishes at the trailing edge.
A more practical foil shape may be obtained by relaxing the requirement of uniform loading near the trailing edge and assuming a vortex distribution that vanishes there. The simplest family of such foils is the NACA
a series, with constant loading from the leading edge downstream over a
distance al, followed by a linear attenuation as shown in figure 5.11. The
corresponding mean-camber lines are shown in figure 5.12, and a = 0.8 is
a common choice for practical purposes. A detailed account of these and
other foil geometries is given by Abbott and von Doenhoff (1959).
5.5 Drag Force on a Two-Dimensional Foil
Thus far in our discussion of two-dimensional steady foils, we have
neglected the drag force parallel to the x-axis. One reason for this omission
is that lifting surfaces are intended to produce lift forces with a minimum
of drag or with a large lift-drag ratio L/D. However, a second reason for
Figure 5.11
Loading of NACA a mean-camber lines.
Lifting Surfaces
185
Figure 5.12
Mean-camber lines for NACA a series.
omitting drag from the preceding analysis is that no drag force acts upon a
steady two-dimensional foil in an inviscid fluid. Viscous drag must be anticipated in a real fluid, and experimental results such as those shown in figure
2.7 confirm this, but the magnitude of the total drag force is comparable
to the skin-friction drag coefficient for small angles of attack. Subsequently,
in dealing with three-dimensional lifting surfaces and with unsteady twodimensional foils, a nonzero drag force will occur in the inviscid theory, but
this cannot be analyzed within the steady two-dimensional theory.
The absence of a drag force, despite the existence of lift, can be verified from the general form of the Kutta-Joukowski theorem (equation 15;
see also problem 3). The most general form of this theorem states that the
hydrodynamic force vector is equal in magnitude to the product ρUΓ, with
Γ the total circulation about the foil; it also states that this force is normal
to the direction of the streaming flow.
A more elementary proof of zero drag follows from an energy analysis.
The velocity field induced by the foil is steady state in a coordinate system
moving with the foil and vanishes at large distances from the foil. The
186
Chapter 5
kinetic energy of the fluid surrounding the foil will move with the foil, and
there can be no net build-up of energy in the fluid downstream. Thus, no
work is done in propelling the foil in a steady-state manner, and there can
be no drag force.
A curious paradox arises in conjunction with the drag force acting on a
flat-plate foil. Given the geometry of this case, it might be expected that the
pressure force acts normal to the plate; in that event the drag force must
equal the product of the lift force and the tangent of the angle of attack.
However, an additional contribution to the hydrodynamic force in this case
results from the infinite velocity at the leading edge and the occurrence of
a corresponding leading-edge suction force. The magnitude of the leadingedge suction force can be calculated from a momentum analysis carried out
locally near the leading edge, and the magnitude of the resulting upstream
force is such that the drag component of the normal force is precisely canceled. Thus, despite the suggestive geometry, the total hydrodynamic drag
force is indeed zero, and the force vector is precisely normal to the direction
of the incident stream.
5.6 Two-Dimensional Source and Vortex Distributions
In the previous sections, vortex distributions have been utilized to represent
the two-dimensional lifting problem; similarly, source distributions can be
used to solve the thickness problem. These two problems are related—both
the source and the vortex are conjugate functions, equal to the real and
imaginary parts of the complex logarithm. This relationship will be pursued
here, not only to derive the corresponding solution of the thickness problem, but also as a necessary first step toward deriving the solution (32) of
the integral equation (30).
Complex variables will be used here, as in section 4.7, with z = x + iy.
The complex potential of a source is given in table 4.1, and that of a vortex
by (23). The combination of these two singularities, situated at the point
(ξ, 0), is
φ + iψ =
1
( q − iγ )log( z − ξ ).
2π
(50)
Here q is the source strength and γ is the vortex strength. Differentiation
with respect to z gives the corresponding complex velocity
Lifting Surfaces
u( z ) − iv( z ) = −
187
1
1
( q − iγ )
.
ξ−z
2π
(51)
If (51) is rewritten in polar coordinates, centered at the singular point (ξ, 0),
then q is the rate of flux of fluid passing radially outward through a circle of
constant radius, and γ is the circulation around the contour.
To represent a two-dimensional hydrofoil, these singularities are distributed continuously along the cut −l/2 < x < l/2 of the x-axis, with an arbitrary source strength q(ξ) and vortex strength γ(ξ). To simplify the following
equations, the coordinates will be nondimensionalized so that l = 2. Then
the complex velocity of this distribution is
1
u( z ) − iv( z ) = −
dξ
1
[ q(ξ ) − iγ (ξ )]
.
ξ−z
2π −∫1
(52)
If the point z = x + iy is close to the boundary y = 0, the values of u and
v on the foil surface can be calculated. For this purpose we let z → x ± i0,
corresponding to the limiting values on the upper and lower surfaces of
the foil. Since there is a pole at the point z = ξ. some care is necessary. The
point z may approach the real axis from above or below if the contour of
integration in (52) is deformed to remain on the original side of the pole,
as shown in figure 5.13. Then, from the calculus of residues, the integral
is equal to a principal-value integral along the real axis, plus or minus one
half of the residue at the singular point ξ = x (see Hildebrand 1976, section
10.15). Thus the limiting values of the velocity components on the upper
or lower surfaces of the cut are given by
1
u± ( x) − iv ± ( x) = −
1
dξ
1
− [ q(ξ ) − iγ (ξ )]
∓ i[ q( x) − iγ ( x)],
ξ−x 2
2π −∫1
Figure 5.13
Equivalent contours of integration as the point z approaches the real axis.
(53)
188
Chapter 5
where −∫ denotes the principal-value integral. To emphasize the importance
of this equation, let us consider the separate cases of a source distribution
(γ = 0) and a vortex distribution (q = 0):
Source Distribution
1
u± ( x) − iv ± ( x) = −
1 q(ξ )dξ 1
−
∓ iq( x),
2π −∫1 ξ − x
2
(54)
Vortex Distribution
1
u± ( x) − iv ± ( x) = −
i γ (ξ )dξ 1
−
∓ γ ( x).
2π −∫1 ξ − x
2
(55)
For the source distribution, separating the real and imaginary parts of
equation (54) gives the resulting velocity components:
1
u± ( x) = −
1 q(ξ )dξ
−
,
2π −∫1 ξ − x
1
v ± ( x) = ± q( x).
2
(56)
(57)
The horizontal velocity component u is even in y, whereas the vertical component v is odd. From the boundary condition (20) for the thickness problem, it follows that the solution for the flow past a thin symmetrical strut,
of thickness t(x) = yu(x) − yl(x), can be obtained directly as a distribution of
sources with strength
q( x) = −Ut ′( x).
(58)
This conclusion is consistent with the result of Green’s theorem noted in
section 4.11.
A simple physical argument can be used to explain (56–58). From symmetry a source at (ξ, 0) on the real axis will induce a flow radially outward
in all directions from this point. Elsewhere on the real axis, the induced
velocity will be horizontal. Thus the only contribution to the vertical velocity on or near the real axis will come from the immediate vicinity of the
source. Near this point, the net outward flux away from the real axis must
equal the source strength, in agreement with (57). On the other hand, the
source will induce a horizontal velocity all along the real axis, of magnitude inversely proportional to the distance x − ξ and thus a distribution of
sources will give a total horizontal velocity consistent with (56).
Lifting Surfaces
189
For the vortex distribution (55), the velocity components on the cut are
given by
1
u± ( x) = ∓ γ ( x),
2
(59)
1
v ± ( x) = −
1 γ (ξ ) dξ
−
.
2π −∫1 ξ − x
(60)
The same pair of equations was derived earlier (29–30) using real variables. The conjugate nature of these results is apparent, relative to (56–57).
In particular, the velocity components (u, v) of a source distribution are
identical to the conjugate pair (v, −u) of the vortex distribution, except for
the change in notation of the singularity strength. Thus, in the lifting problem, the local vortex strength γ can be related to the jump in tangential
velocity at the same point on the mean-camber line, but not to the normal
velocity.
5.7 Singular Integral Equations
To solve the lifting problem with prescribed foil geometry or a given v(x) on
the cut, it is necessary to determine the vortex strength γ from the integral
equation (60). Physically, each vortex element along the cut contributes to
the net vertical velocity at any other point on the cut, and the problem is to
find the appropriate distribution of vortices that induce a prescribed value
of v(x). Equation (60) is a singular integral equation because the kernel, or
known part of the integrand, is singular.
The literature on the solution of singular integral equations is extensive;
see, for example, the monograph of Muskhelishvili (1953) and a section
in the text by Carrier, Krook, and Pearson (1966). Here we shall construct
the solution in a manner that exploits our physical understanding of the
thickness problem and the conjugate nature of the source and vortex
distributions.
Figure 5.14 indicates the linearized boundary-value problem appropriate to each physical situation. The vertical velocity component v+ denotes
the normal velocity on the upper surface of each body, which is assumed
known from the boundary conditions (20–21). From the appropriate symmetries, v(x, y) is an odd function of y in the thickness problem, and hence
in this case v(x, 0) = 0 on the real axis outside the cut. Conversely, for the
190
Chapter 5
Figure 5.14
Boundary values on the cut real axis in the thickness and lifting problems.
lifting problem, u(x, y) is an odd function of y which must vanish on the
real axis outside the cut.
Now let us consider the upper half of the two sketches in figure 5.14.
Apparently, the source strength can be found directly in the first case
because the vertical velocity is given along the entire upper side of the real
axis, from minus infinity to plus infinity, and equation (57) gives an appropriate source distribution that will provide this velocity distribution. On the
other hand, the lifting problem is a mixed boundary-value problem since
the vertical velocity is prescribed only on the cut, with the horizontal velocity prescribed elsewhere on the real axis. However, if the complex velocity
u − iv in the lifting problem is multiplied by a suitably chosen analytic function, such as (1 − z2)1/2, which is real on the cut and pure imaginary on the
remainder of the real axis, then the product will be a new analytic function
whose imaginary part is known all along the real axis. The square-root function (1 − z2)1/2 is clearly not a unique choice, but it is at least a possible one,
and the resulting lack of uniqueness must be faced later.
The procedure for converting the lifting problem into an alternative
mathematical problem of the same form as the thickness problem is then
as follows. A new pseudo-velocity function is defined as
u − iv = ( u − iv )(1 − z 2 )1 2 .
(61)
This new function must satisfy the following boundary conditions, on the
upper side of the real axis:
v = v + (1 − x2 )1 2 ,
x < 1,
(62)
v = 0,
x > 1.
(63)
Lifting Surfaces
191
Here the second condition (63) results from the fact that u = 0 on the real
axis outside the cut. Mathematically, this problem is indistinguishable from
the upper half of the thickness problem, except for the modification in
the vertical velocity due to the square root (62). Thus, the pseudo-velocity
function (61) can be generated with a source distribution, of strength given
by (57), or by
q( x) = 2v( x) = 2v + ( x)(1 − x2 )1 2 ,
q( x) = 0,
x < 1,
x > 1.
(64)
(65)
From (52) it follows that in the upper half-plane,
u ( z ) − iv( z ) = −
1
1 v + (ξ )(1 − ξ 2 )1 2
dξ.
π −∫1
ξ−z
(66)
Thus, from (61), the original velocity function in the upper half-plane must
be of the form
u − iv = −
1
1
π (1 − z 2 )1 2
1
v + (ξ )(1 − ξ 2 )1 2
dξ.
ξ−z
−1
∫
(67)
The integral equation (60) for the vortex strength γ can then be solved
by using (67) to determine the horizontal velocity component on the upper
side of the cut,
1
u( x, 0 + ) = −
v (ξ )(1 − ξ 2 )1 2
1
1
− +
dξ.
2 12 ∫
π (1 − x ) −1
ξ−x
(68)
From (59), the solution for γ is
γ ( x) =
2
1
π (1 − x2 )1 2
1
v + (ξ )(1 − ξ 2 )1 2
dξ.
ξ−x
−1
∫
(69)
One deficiency of (69) is the presence of square-root singularities at both
the leading and trailing edges. Obviously, it would be coincidental if the
Kutta condition at the trailing edge was satisfied at this stage, for we have
proceeded so far without regard for the singularities at the two ends of the
cut. This problem is related to the nonunique choice of a square-root function used to generate the pseudo-velocity function (61). If we had anticipated the desire to satisfy the Kutta condition at the trailing edge, we might
have used the alternative function
192
Chapter 5
1 − z1 2
,
u − iv = ( u − iv ) 
1+ z
(70)
instead of (61), and thus obtained the final solution, corresponding to (69),
as
γ ( x) =
1
2  1 + x1 2  1 − ξ 
−
π  1 − x  −∫1  1 + ξ 
12
v + (ξ )
dξ.
ξ−x
(71)
In fact, (71) is the desired solution, singular only at the leading edge. At
this point, however, we may feel some concern about the lack of a unique
answer to the original integral equation. In essence, (69) and (71) are two
particular solutions of the integral equation (60), and in general a homogeneous solution must be added in a manner analogous to that employed in
solving differential equations.
The homogeneous solution of (60) satisfies this integral equation with
the left side set equal to zero. To find this function, we could guess, look
through a table of integrals, or simply accept that the answer is the inverse
square-root function [1 − x2]−1/2. To verify this more systematically, let us
find the difference between the two particular solutions (69) and (71). After
some algebra, the result is
1
v (ξ )
2
1
 ξ − x
− + (1 − ξ 2 )1 2 
dξ
 1 + ξ 
π (1 − x2 )1 2 −∫1 ξ − x
1
=
2
1
1− ξ
− v + (ξ ) 
 1 + ξ 
π (1 − x2 )1 2 −∫1
12
dξ =
C
2
,
π (1 − x2 )1 2
(72)
where C is a constant, equal to the last integral. Thus the inverse squareroot function on the right side of (72) is a homogeneous solution of the
integral equation (60).
To relate the constant C in (72) to the total circulation, we write the general solution for γ in the form
γ ( x) =

 1 v + (ξ )(1 − ξ 2 )1 2
2
1
−
dξ + C  ,
2 1 2 ∫
π (1 − x )  −1
ξ−x

(73)
where C is now an arbitrary constant. Integrating (73) between (−1, 1) gives
the total circulation,
1
Γ = ∫ γ ( x) dx.
−1
(74)
Lifting Surfaces
193
The right side of (73) may be integrated as follows:
Γ=
1
1
1
=
1
2
dx
v (ξ )(1 − ξ 2 )1 2
2C
dx
− +
dξ +
π −∫1 (1 − x2 )1 2 −∫1
ξ−x
π −∫1 (1 − x2 )1 2
1
dx
2
dξ v + (ξ )(1 − ξ 2 )1 2 ∫−
+ 2C.
π −∫1
(ξ − x)(1 − x2 )1 2
−1
(75)
Since the inverse square-root is the homogeneous solution of the integral
equation (60), the last integral in (75) must be zero, and thus
C=
1
Γ.
2
(76)
Now the Kutta condition can be imposed by choosing the constant
C=
1
Γ so that the factor in brackets in (73) is zero at x = −1, canceling the
2
square-root infinity at this point. Thus,
1
C=
1
1
v (ξ )(1 − ξ 2 )1 2
1− ξ
Γ = −∫ +
d ξ = − ∫ v + (ξ ) 
 1 + ξ 
2
ξ +1
−1
−1
12
dξ ,
(77)
and this is consistent with (72). In this manner the particular solution (71)
can be obtained by a systematic analysis from the general solution (73),
after imposing the Kutta condition at the trailing edge.
This completes our solution of the integral equation (60) and provides a
derivation of (32) and (34). We can go one step further, however, and show
how the resulting definite integrals can be evaluated. One type of definite
integral required to evaluate the lift and moment coefficients is given by
1
In =
−1
π
xn dx
= ∫ (cos θ )n dθ .
2 12
)
0
∫ (1 − x
(78)
Note that (36) and (38) can be expressed as combinations of these integrals,
whenever the slope of the mean-camber line is a polynomial in x. The integrals in (78) are elementary integrals; when n is odd they equal zero, and
when n is even
In = π
1 ⋅ 3 ⋅ 5 ⋅ ... ⋅ ( n − 1)
.
2 ⋅ 4 ⋅ 6 ⋅ ... ⋅ ( n )
(79)
The corresponding integrals, which must be evaluated to find the vortex
strength, are of the form
1
ξ n dξ
.
(ξ − x)(1 − ξ 2 )1 2
−1
H n ( x) = ∫−
(80)
194
Chapter 5
These are not elementary, but they can be related to the integrals in (78).
First we recall that, since (l − ξ2)−1/2 is a homogeneous solution of the integral equation (60),
1
dξ
= 0.
ξ
(
−
)(
1 − ξ 2 )1 2
x
−1
H 0 ( x) = ∫−
(81)
For n ≥ 1 in (80), a recursion formula can be derived as follows:
1
[(ξ − x)ξ n −1 + xξ n −1 ]dξ
= I n −1 + xH n −1( x).
(ξ − x)(1 − ξ 2 )1 2
−1
H n ( x) = ∫−
(82)
In particular, it follows from (79) and (81) that
H1( x ) = π ,
(83)
H 2 ( x ) = π x,
(84)
and, more generally, Hn(x) is a polynomial of degree n − l.
The integrals defined by (80) are known more generally as Hilbert transforms; so far we have considered them only on the cut −1 < x < 1. This is the
region of principal interest for steady two-dimensional hydrofoils, but in
our subsequent treatments of cavity flows and unsteady hydrofoils, where
singular effects exist downstream of the foil, it will be necessary to evaluate the resulting Hilbert transforms for values of x exterior to the cut. This
generalization is accomplished most easily by replacing x by the complex
variable z and using analytic continuation to deduce the values of (80) off
the cut. The recursion formula (82) remains valid in this case, but the other
results must be modified.
Since (82) is valid, it is necessary to consider only the generalization of
(81) which, for complex z, takes the form
1
H0(z) =
dξ
∫ (ξ − z )(1 − ξ
−1
2 12
)
.
(85)
Recalling the discussion following figure 5.13, the limiting behavior of (85)
on the cut is given by
H 0 ( z ) → ± π i(1 − x2 )−1 2 ,
(86)
for z → x ± i0, where the real part, H0(x), vanishes by virtue of (81). Since
(85) is an analytic function of z, it follows that
H 0 ( z ) = −π ( z 2 − 1)−1 2 ,
(87)
Lifting Surfaces
195
since this is the only analytic function consistent with (86). Here the
branch of the square root is prescribed so that H0(z) ≃ −π/z at infinity, a
result which can be derived independently from (85) using (79). For z = x
outside the cut, the desired result follows in the form
x > 1

 x < −1
H 0 ( x) = ∓ π ( x2 − 1)−1 2 ,
(88)
Analogous expressions for Hn(x) can be deduced from (82).
The family of integrals (80) can be expressed in an alternative and particularly useful form, by making the substitutions x = cos χ. ξ = cos θ. Since
cosnθ can be expressed as a finite Fourier series involving typical terms of
the form cos mθ, m ≤ n, we consider the family
π
cos nθ dθ
;
θ − cos χ
cos
0
Gn ( χ ) = ∫−
(89)
these are known as Glauert integrals. From (81), (83), and (84) it is apparent
that the first three members are
G0 = 0,
(90)
G1 = π ,
(91)
G2 = 2π cos χ .
(92)
More generally,
Gn = π
sin n χ
.
sin χ
(93)
This simple result can be confirmed by using the addition formula
cos( n + 1)θ + cos( n − 1)θ = 2 cos nθ cos θ ,
(94)
to derive from (89) the recursion relation
π
Gn +1 + Gn −1 − 2 cos χ Gn = ∫ cos nθ dθ = 0,
n ≥ 1.
(95)
0
Using the addition formula
sin( n + 1) χ + sin( n − 1) χ − 2 cos χ sin n χ = 0,
(96)
we find that (93) satisfies (95). Since it is consistent with (90–92), it must
hold in general.
196
Chapter 5
The last results suggest a Fourier series approach which leads to particularly simple results. In dimensional coordinates, with x = 21 l cos θ , the
vortex strength can be expanded in the following trigonometric series:
∞
1


γ = 2U  A0 cot  θ  + ∑ An sin nθ  .


2


n =1
(97)
This series has been chosen so that each term satisfies the Kutta condition
at the trailing edge θ = π. The first term is proportional to the flat-plate load
distribution (42), and represents the flat plate if A0 = α. The remaining terms
in (97) vanish at the leading edge, and this infinite series is a complete Fourier series representation for any reasonable vortex distribution vanishing
at the leading and trailing edges; thus, it follows that (97) can be used to
represent practically any realistic vortex distribution, with the leading-edge
singularity absorbed in the first term.
Using equation (60), the vertical velocity is given by
v=−
π
1 γ (θ1 )sin θ1 dθ1
−
.
2π ∫0 cos θ1 − cos θ
(98)
Substitute (97) for γ, and use (89), (93) and the trigonometric relations
cos
cos
sin
sin ( n − 1)θ − sin ( n + 1)θ = ± 2 sin θ cos nθ
(99)
and
1 + cos θ
1
cot  θ  =
.
2 
sin θ
(100)
Then it follows from (98) that
∞
v = −UA0 + U ∑ An cos nθ .
(101)
1
Alternatively, from the boundary condition (22),
∞
dη
= A0 − ∑ An cos nθ .
dx
1
(102)
Hence the coefficients An can be related to the slope of the mean-camber
line, from the usual orthogonality property of Fourier coefficients, by
means of the equations
Lifting Surfaces
A0 =
197
π
1 dη
dθ ,
π ∫0 dx
An = −
(103)
π
2 dη
cos nθ dθ .
π ∫0 dx
(104)
Finally, the lift coefficient is given by
CL =
π
L
1
= ∫ γ sin θ dθ = π (2 A0 + A1 ),
2
½ρU l U 0
(105)
and the coefficient of the moment about the mid-chord position is
CM =
π
M
1
1
π
=
γ cos θ sin θ dθ =  A0 + A2  .
½ρU 2l 2 2U ∫0
2
2 
(106)
The results (103–106) are equivalent to (36) and (38).
5.8 Three-Dimensional Vortices
The preceding analysis of two-dimensional lifting surfaces can be extended
to three dimensions, using distributions of vortices in the plane y = 0. For
this purpose the properties of three-dimensional vortices must be developed.
A three-dimensional line vortex, or vortex filament, is a singularity situated along an arbitrary curve in space. This singularity has the same local
properties as a two-dimensional point vortex—the flow very close to the
filament appears locally as a two-dimensional vortex when observed in a
plane normal to the filament (see figure 5.15). A smoke ring is a familiar
example of a vortex filament that occupies a closed circular contour.
Figure 5.15
Three-dimensional vortex filament.
198
Chapter 5
The flow must be irrotational throughout the fluid region, excluding the
line on which the vortex filament is situated. Thus, the circulation around
any closed path that does not include the vortex must be zero, while the
circulation around a small, closed path about the vortex must be equal to
the circulation of the vortex at that point. To see that this circulation is
the same for all points along the filament, define a closed contour C1 + C2
+ C3 + C4 to form the boundary of a tube surrounding the vortex filament,
as shown in figure 5.16. Since C1 + C2 + C3 + C4 is a closed curve, which
does not include the vortex, the total circulation around this contour must
vanish. The contributions from C2 and C4 cancel each other, since these
two contours are practically identical except for the change in the positive
direction of integration. Thus the contributions from C1 and C3 must be
equal and opposite, or
∫ V ⋅ dl + ∫ V ⋅ dl = 0.
C1
(107)
C3
Figure 5.16 shows that the contours C1 and C3 are opposite in their
sense of rotation about the vortex filament, and if the first integral in (107)
defines the circulation Γ1. the second integral will be equal to −Γ3 and vice
versa. Thus Γ1 = Γ3, and the circulation of a vortex filament must be constant along its length. As a consequence of this result, the circulation is
conserved, and a vortex filament cannot end within the fluid. A vortex filament can terminate at the boundaries of the fluid domain, but otherwise it
must form a closed curve, as shown in figure 5.15.
The velocity field induced by a vortex filament can be deduced from
the two-dimensional results (24–25). For motion in the x-y plane due to a
point vortex at the origin of strength Γ, the velocity components can be
written as
Figure 5.16
Contour used to show constancy of circulation along a vortex filament.
Lifting Surfaces
u=−
199
Γy
Γ ∂
=−
log r ,
2π ( x2 + y 2 )
2π ∂y
(108)
Γx
Γ ∂
log r .
=−
2
2
2π ( x + y )
2π ∂x
(109)
and
v=−
Thus, the velocity vector in this case can be written as
V=
−Γ
∇ × k log r .
2π
(110)
Here k denotes the unit vector in the z-direction, parallel to the axis of the
vortex.
To pass from two to three dimensions, we replace the two-dimensional
source potential log r by the potential of a line of three-dimensional sources,
of strength m, initially along the z-axis:
V =∇×
=
∞
mk
dz
∫ ( x2 + y 2 + z 2 )1 2
4π −∞
∞
−m
( iy − jx) dz
−m( iy − jx)
∫ ( x2 + y 2 + z 2 )3 2 = 2π ( x2 + y 2 ) .
4π −∞
(111)
The desired two-dimensional velocity components (110) will result from
a line distribution of three-dimensional sources normal to the x-y plane,
provided the source strength is m = Γ.
Now we rewrite the last line integral in (111) as
V=−
Γ
4π
∞
∫
−∞
R × k dz
,
R3
(112)
where R is the position vector xi + yj + zk.
Equation (112) gives a form of the two-dimensional velocity field resulting from a line vortex along the z-axis, which can be generalized to an
arbitrary three-dimensional vortex filament of strength Γ. Let the vortex
filament be situated on a contour C, with dl the differential element of
integration along C as in figure 5.17. Then by analogy with (112) we can
define the velocity
V=−
Γ R × dl
,
4π ∫C R3
(113)
200
Chapter 5
Figure 5.17
Definition sketch for equation (113), showing the position vector R and differential
line element dl along a vortex filament situated on the contour C.
where R is now the vector to the point (x, y, z) from each point along the
curve of integration, as defined in figure 5.17. This more general integral
defines a velocity field that is irrotational and divergenceless except on the
contour C. Near this contour, it possesses the same singularity as the twodimensional vortex (112), since locally the curve approximates a straight
line and near the curve the integral will be dominated by the value of the
integrand as R → 0.
Equation (113) is, therefore, the velocity field induced by a threedimensional vortex filament of strength Γ situated on the curve C. This
result, well known in electromagnetic theory as the law of Biot-Savart,
relates the magnetic field to the current in a conducting filament. We have
derived it intuitively by guessing and then verifying the final answer. A
more systematic derivation, in terms of an arbitrary volume distribution
of vorticity reduced ultimately to a line vortex, can be found in Batchelor
(1967). An extensive analysis of three-dimensional vortex fields is given by
Brard (1972).
5.9 Three-Dimensional Planar Lifting Surfaces
For three-dimensional lifting surfaces and hydrofoils, the powerful method
of complex function theory must be abandoned; the real variable z and
dummy coordinate ζ are used to denote the coordinate along the span of
the foil. However, the thin-wing approximation can be retained if the slope
of the foil is sufficiently small over its entire surface (except at the leading
edge!). A linearized problem can be formulated in exactly the same manner
as for the two-dimensional case in section 5.2. The thickness problem can
be solved again in terms of a suitable source distribution, and the lifting
Lifting Surfaces
201
problem can be reduced to one involving a mean-camber surface of zero
thickness where there is a specified normal velocity
v( x, 0, z ) = −U
∂
η( x, z ).
∂x
(114)
This boundary condition must be supplemented by the Kutta condition
requiring finite velocity all along the trailing edge.
As in the two-dimensional case, a solution of the boundary condition
(114) can be constructed by a suitable distribution of vortices in the plane
y = 0. In general, such a distribution would be comprised of line vortices of
varying strengths and varying orientation angles. In other words, γ in this
case is a vector situated in the plane y = 0. This vector can be decomposed
into a bound component, parallel to the z-axis, and a free component, parallel to the x-axis. Thus it is necessary to consider only these two specific orientations and the induced velocities corresponding to each. In section 5.8
we showed that the limiting behavior of a line vortex very close to its axis
is identical to that of a two-dimensional point vortex of the same strength.
Thus the results derived in section 5.2, for a distribution of two-dimensional
vortices along the x-axis, will apply locally for the bound vortex distribution in three dimensions. In particular, a bound vortex distribution of density γB(x, z) will induce a discontinuity in the chordwise component of the
fluid velocity, across the plane y = 0, in accordance with the relation
1
u( x, ± 0, z ) = ∓ γ B ( x, z ).
2
(115)
Similarly, a free vortex distribution of density γF(x, z) will induce a discontinuity in the spanwise component across y = 0 given by
1
w( x, ± 0, z ) = ± γ F ( x, z ).
2
(116)
Note that the right-hand rule applies here; the circulation of the bound
vortices is defined as positive about the z-axis and that of the free vortices
positive about the x-axis.
These planar distributions of vortices are known generally as vortex
sheets, and we may refer separately to the bound vortex sheet and the free
vortex sheet. These are not independent, however, since the velocity field
adjacent to the sheets must be irrotational. Using (115) and (116) to compute the y-component of the vorticity adjacent to the foil, we get
202
Chapter 5
∂u ∂w
−
= 0,
∂z ∂x
(117)
or
∂γ B ∂γ F
−
= 0.
∂z
∂x
(118)
This is essentially the extension to a vortex sheet of the result in section
5.8 for a vortex filament. Each element of bound vorticity cannot change
in the z-direction, unless the change in its circulation is balanced by a corresponding change in the free vorticity.
The linearized pressure induced by these vortex sheets can be computed
from (14), since the additional velocity component w does not give a linear
contribution. Thus, on the upper or lower surface of the vortex sheet,
p − p∞ ρuU = ∓
1
ρU γ B ( x, z ).
2
(119)
Therefore, the pressure jump across the sheet is proportional to γB. and the
pressure associated with the free vortex sheet is continuous.
The absence of a pressure jump for the free vortex sheet is an important
feature of this component of the total vortex distribution, which is responsible for its name. A free vortex sheet, composed of vortex elements parallel
to the free stream, can exist in a state of dynamic equilibrium within the
free stream, whereas bound vortices with axes perpendicular to the free
stream induce a pressure jump that requires an external balancing force,
that is, the force acting on the lifting surface.
Now we apply these results to the lifting-surface problem by distributing
sheets of bound and trailing vorticity on the plane y = 0 throughout the
projection of the lifting surface planform. Integration of (119) along the
chord, at a section of constant z, gives the sectional lift force
L( z ) = ρU
xL ( z )
∫
γ B ( x, z ) dx = ρU Γ ( z ).
(120)
xT ( z )
Here Γ(z) is the total circulation of the section, and the limits of integration
denote the x-coordinates of the trailing and leading edges. A second integration in the spanwise direction gives the total lift
L = ρU
s /2
∫
− s /2
Γ ( z ) dz.
(121)
Lifting Surfaces
203
Ultimately this equation can be used to compute the total lift force, but
first the appropriate distributions of bound and trailing vortices must be
explored.
Integrating (118) along the chord gives the following equation for continuity of vorticity across the section:
xL
 ∂γ B + ∂γ F  dx = 0,
∂z
∂x 
xT
∫
(122)
or
Γ ′( z ) + [γ F ( x, z )]xxTL = 0.
(123)
Thus, any change in the total circulation along the span of the lifting
surface must be reflected in a corresponding jump of free vortex density
across the section. The circulation Γ clearly must vary with z, especially
1
s where Γ changes from a nonzero value on the
2
lifting surface to zero outside the tips. But where is the resulting free vorticnear the tips z = ±
ity to go? It too must be conserved and cannot end abruptly at the leading
or trailing edges. This apparent dilemma can be resolved by assuming that
the lifting surface advances into a uniform and undisturbed fluid, but that
it leaves behind a thin wake region of free, or trailing, vorticity. The wake
extends to infinity downstream, or alternatively to the downstream position of the starting vortex. Since this starting vortex is equal and opposite
to Γ(z), the system of bound and trailing vortices is rejoined downstream
in a closed loop.
The existence of a trailing vortex sheet downstream of the lifting surface
is physically permissible since free vortices can exist in a state of dynamic
equilibrium with the fluid. With the free vorticity set equal to zero upstream
of the leading edge and equal to the trailing vorticity γT(z) at the trailing
edge, (123) determines the value of the trailing vorticity as
γ T ( z ) = Γ ′( z ).
(124)
We emphasize that γT is independent of the x-coordinate, by virtue of (118)
and because there can be no bound vorticity in the wake.
These vortex distributions are sketched in figure 5.18. Here the lifting
surface, (or, strictly, its projection on the plane y = 0) contains a distribution
204
Chapter 5
Figure 5.18
Sketch showing examples of the discrete vortices, which make up the continuous
vortex sheets on the foil and downstream in the wake.
Figure 5.19
Horseshoe vortex of strength γ.
of bound and free vortices; the latter is joined at the trailing edge to a sheet
of trailing vortices that extend downstream.
A particularly simple case is the lifting surface of small chord length, or
large aspect ratio, and constant sectional loading. The bound vortex distribution is then constant along the span and confined to a small chord, but
at the tips, it must be connected to two discrete trailing vortices as shown
in figure 5.19. Note that (124) still holds in this case, but the derivative is
equal to zero except at the tips,, where it is infinite, and hence γT is given
by a pair of delta functions.
Lifting Surfaces
205
The line vortex shown in figure 5.19 is a horseshoe vortex, which itself
satisfies the requirement of conservation of circulation. A system of these
may be superposed to represent a more general lifting surface. Horseshoe
vortices of equal span may be distributed in the chordwise direction to
make a foil of rectangular planform and constant sectional loading γ, as
shown in figure 5.20. However, while the loading in this case is constant
along the span, the downwash, or the velocity −v (x, 0, z) induced on the
planform by this vortex distribution, must depend on z, and a pronounced
local downwash will exist near the tips induced by the trailing vortices.
Thus, from the boundary condition (114), the geometry of the foil in this
case is not independent of span wise position and, indeed, must be physically unrealistic near the tips.
Another special case results by distributing horseshoe vortices of varying
span along the same chordwise position, as shown in figure 5.21. Here the
individual elements combine to form a single bound vortex Γ (z), and the
Figure 5.20
Lifting surface with uniform spanwise loading, composed of horseshoe vortices of
equal span.
Figure 5.21
Lifting-line, composed of horseshoe vortices with coincident bound elements but
different spans.
206
Chapter 5
trailing vorticity is consistent with (124). This case is of practical importance for lifting surfaces of large aspect ratio, where the chord length is
small and hence the chordwise position of the vortex elements is unimportant. The resulting approximation, for lifting surfaces of large aspect ratio, is
the lifting-line theory that will be treated in section 5.11.
The trailing vortex sheet, which is an inevitable consequence of threedimensional effects, is detrimental to the performance of a lifting surface.
The effects are twofold: the lift is reduced and the drag increased. For the
lift, figures 5.19–5.21 show that the sign of the vortices, associated with a
positive lift force in the y-direction, induces a downward component of vertical velocity along the foil. This vertical downwash can be superposed with
the incident stream velocity vector, resulting in a decreased effective angle
of attack, by comparison with the two-dimensional case without downwash. Thus the total lift force L is less than would be expected from a twodimensional analysis at each section.
5.10 Induced Drag
In three dimensions a drag force exists because the trailing vortex sheet
increases in length, at a rate proportional to U, with a resulting increase in
the total kinetic energy of the fluid at a similar rate. Since this energy must
be supplied by work done to overcome the drag force, the induced drag D
will be equal to the kinetic energy of a slice of fluid far downstream, the
slice being of unit width along the x-axis.
Alternatively, from the general force equation (4.90), the drag component is given by the integral
D=
 ∂φ 2  ∂φ  2
1
∂φ 2 
ρ ∫∫   −   −    nx dy dz.
 ∂z  
 ∂y 
2 Sc  ∂x 

(125)
Here the control surface has been taken as a pair of planes x = constant
upstream and downstream of the foil, and the remaining surface between
these at infinity can be neglected since the perturbation vanishes sufficiently
rapidly as y2 + z2 → ∞. Indeed, the same must be true upstream but not downstream where the trailing vortex sheet persists. Since the velocity induced by
the vortex sheet far downstream is independent of x, it follows that
Lifting Surfaces
D=
1
ρ
2 ∫
−∞
∫
∞
207
  ∂φ  2  ∂φ  2 
  +    dy dz ,
∂z 
 ∂y 
(126)
where the integrand is to be evaluated for x → −∞. This integral is the twodimensional kinetic energy in a Trefftz plane x = constant, far downstream.
The surface integral in (126) can be replaced by a line integral around
1
the trailing vortex sheet, or the cut z < s , by means of the divergence
2
theorem. Since ▽2ϕ = 0, it follows that
1
1
ρ ∇φ ⋅ ∇φ dy dz = ρ ∫∫ ∇ ⋅ (φ∇φ ) dy dz
2 ∫∫
2
1
∂φ
= ρ
φ
dl
2 ∫ ∂n
D=
(127)
s2
=−
1
∂φ
dz.
ρ [φ ]+−
∂y
2 − ∫s 2
Note that the contour integral at large radial distance (y2 + z2)1/2 in the
Trefftz plane is zero, since the velocity components v and w vanish at a sufficient rate far from the trailing vortex sheet. Also, in the final step of (127),
we have used the fact that ∂ϕ/∂y = ± ∂ϕ/∂n is continuous across the wake,
whereas the potential is discontinuous.
The jump in the potential can be related to the bound vorticity by integrating the pressure jump with respect to x and noting that the pressure
jump across the wake must vanish. Thus
L
L
∂φ +
∂φ +
Γ ( z ) = − ∫   dx = − ∫   dx = [φ ]+−
 ∂x  −
 ∂x  −
xT
−∞
x
x
x =−∞
,
(128)
since the jump in the potential must vanish at the leading edge. Substituting (128) into (127), we obtain the drag formula
s2
D=−
1
∂φ
ρ Γ ( z ) dz ,
2 − ∫s 2
∂y
(129)
where the integration is taken across the vortex wake far behind the foil.
The vertical velocity component ∂ϕ/∂y, far downstream on the trailing
vortex sheet, can be computed from the equation (60) for a two-dimensional
vortex distribution. With a suitable change in the coordinate system and
using (124) to evaluate the trailing vortex density, we get
208
Chapter 5
s2
d Γ (ζ ) dζ
∂φ
1
−
=
.
∂y 2π − ∫s 2 dζ ζ − z
(130)
(The change of sign from (60) is necessitated by the transformation from
the x-y plane to the y-z plane, while defining the vorticity consistently in
terms of the right-hand rule.) With this result, (129) may be replaced by a
formula involving only the bound circulation:
D=
ρ
4π
s2
∫
s2
−s 2
Γ ( z ) dz ∫−
−s 2
d Γ (ζ ) dζ
.
dζ z − ζ
(131)
The lift force (121) can be derived by a similar analysis of the flow in the
Trefftz plane. Thus, starting with the y-component of (4.90), and including only the linear term, we get the following expression corresponding to
(126):
L = − ρU ∫
∞
∫
s2
∂φ / ∂y dy dz = ρU
∫
[φ ]+− dz ,
(132)
−s 2
−∞
and (121) follows from (128).
For a given spanwise distribution of circulation Γ(z), equations (121) and
(131) can be used to compute the total lift and drag forces. Since the lift is
linearly proportional to Γ, it follows from (131) that the induced drag will
be proportional to the square of the lift and will depend quadratically on
the angle of attack and camber. Thus, the lift-drag ratio L/D is inversely proportional to the angle of attack and camber, and hence L/D ≫ 1 if the foil
satisfies the original assumptions of the linear theory. (At this point it may
be a relief to note that the lift-drag ratio is large, since that is presumably
the raison d'être for the lifting surface!) That the lift-drag ratio is inversely
proportional to the angle of attack is another reason for keeping this angle
small, in addition to the objectives of avoiding stall, cavitation, and nonlinear effects.
The lift-drag ratio is not a satisfactory figure of merit since this can be
made arbitrarily large by reducing the angle of attack and resulting lift
force. A more appropriate parameter is the nondimensional ratio
K=
=
π ACD π
D
= ρU 2 s2
2
CL 2
L2
s2 ∫ Γ ( z )∫ [ Γ ′(ζ ) / ( z − ζ )] dζ dz
.
2
8
Γ ( z )dz
(∫
)
(133)
Lifting Surfaces
209
Here A is the aspect ratio, as defined by (1);
CL =
L
½ρSU 2
(134)
D
½ρSU 2
(135)
And
CD =
are the three-dimensional lift and drag coefficients based on the planform
area S, and the limits of the integrals in (133) are ± 21 s.
With the physically relevant assumptions that the total circulation Γ(z)
and its derivative Γ′(z) are continuous along the span, and that Γ vanishes
at the tips z = ± 21 s, this function can be expanded in a Fourier series of the
form
∞
1
Γ  s cos θ  = 2Us∑ an sin nθ ,
2

n =1
where z =
(136)
1
s cos θ . Using (121) and (134), the lift coefficient is given by
2
s2
CL =
2
Γ ( z ) dz = π Aa1 ,
US − ∫s 2
(137)
where the aspect ratio A is defined by (1). The drag coefficient (135) can
be computed in a similar fashion from (131) by using the Glauert integrals
(89), and it follows that
s2
CD =
s2
∞
1
d Γ (ζ ) dζ
−
Γ
π
(
)
=
z
dz
A
nan 2 .
∑
∫
∫
2πU 2 S − s 2
dζ z − ζ
n =1
−s 2
(138)
In deriving (137–138), the orthogonality relation
π
π / 2, m = n,
m ≠ n,
∫ sin mθ sin nθ dθ = 0,
0
has been used, with m and n positive integers.
Since the lift coefficient (137) depends only on the Fourier coefficient
a1. whereas the drag coefficient (138) is a positive-definite sum over all an
the optimum spanwise distribution is elliptical, with an = 0 for n ≥ 2; thus
210
Chapter 5
Γ ( z ) ∝ ( s2 4 − z 2 )1 2 .
(139)
In this case the figure of merit K defined by (133) is equal to 1.0. For any
other spanwise loading, K will be greater than unity, and the lifting surface
will be less efficient.
5.11 Lifting-Line Theory
The appropriate representation of a three-dimensional lifting surface, in
terms of planar vortex sheets, has been outlined in section 5.9. A quantitative solution for the vortex densities requires a surface-integral representation for the downwash velocity on the foil, in terms of these vortex
densities, as in the two-dimensional case (30). For a prescribed foil geometry, the boundary condition (114) then yields an integral equation for the
unknown vortex density. However, the unknown γ(x, z) is now a function
of two coordinates, and the corresponding integral equation is a surface
integral. Closed-form solutions corresponding to those derived in section
5.7 cannot be derived, and further progress in the three-dimensional case
requires additional approximations or a numerical solution. This topic is
covered by Robinson and Laurmann (1956) and Thwaites (1960).
Here we shall discuss an approximate technique that adds considerable
insight to the understanding of three-dimensional lifting surfaces and permits relatively simple calculations to be made for hydrofoils of large aspect
ratio. A complementary approximation for small aspect ratios will be given
in chapter 7, as a special case of slender-body theory.
If the aspect ratio is very large, the obvious approximation is a twodimensional strip theory, with each section treated as a two-dimensional
foil with the same geometry and angle of attack. This approach is limited,
however, since trailing vorticity must inevitably be shed and will affect
both the lift and drag forces on the foil. To account for these effects appropriately for large aspect ratios, Prandtl developed the lifting-line theory.
With boundary-layer theory, this must be regarded as an outstanding example of the use of asymptotic approximations to simplify a more complicated
problem. The lifting-line approach of Prandtl can be regarded as a second
approximation, correcting the strip-theory approach, in the same sense
that boundary-layer theory corrects the assumption of inviscid flow.
Lifting Surfaces
211
If the span of the foil is much larger than the chord length, the flow can
be described as it would appear to both an observer in the inner region, with
length scales comparable to the chord, and an observer in the outer region,
with length scales comparable to the span. The corresponding domains are
essentially as sketched in figures 5.2 and 5.21, respectively. Thus, the inner
solution appears to be two dimensional, with an appropriate section geometry for the foil and a corresponding inflow velocity and angle of attack.
From the standpoint of this inner flow, the three-dimensional effects are
slowly varying on the scale of the span, and hence locally constant. This
is not to suggest that three-dimensional effects are negligible in the inner
region; but being locally constant, these can be allowed for by suitably
correcting the inflow velocity vector. Thus, a two-dimensional problem is
anticipated for the inner flow, identical to that treated in sections 5.2 to 5.4,
but modified in the second approximation by an inflow velocity and direction unknown in advance. Once these are given, the inner solution can be
obtained in terms of the two-dimensional foil characteristics, notably the
total circulation Γ(z) at each section.
Next we consider the corresponding outer solution, as sketched in figure 5.21. Here one is too far away to be concerned about details of the foil
geometry, such as its local camber and angle of attack. Instead, the important features are the presence of a bound vortex Γ(z) and corresponding
trailing vortices with density Γ'′(z), which govern the outer flow. The trailing vortex sheet, which is absent from the two-dimensional inner solution,
induces a significant three-dimensional effect on the inner flow, notably
the downwash velocity. To account for this effect, we must calculate the
downwash velocity induced by the trailing vortex sheet.
The problem is simplified considerably by restricting our attention to
lifting surfaces, as shown in figure 5.21, where there is no sweepback angle.
In the outer view of this problem, the foil occupies a straight line or a finite
segment ( − 21 s, 21 s ) of the z-axis. The trailing vortices are situated in the
portion of the plane y = 0 downstream of this segment, and each individual
trailing vortex element coincides with a semi-infinite line −∞ < x < 0, y =
0, z = ζ. The computation of the induced velocity at points on the z-axis is
facilitated by noting from symmetry that a semi-infinite vortex filament,
extending from x = −∞ upstream to the plane x = 0, will induce in this plane
precisely half of the velocity that would result from an infinite line vortex
212
Chapter 5
extending from −∞ to +∞ along the same line. The latter problem is two
dimensional, and thus the vertical component of the velocity induced by
the trailing vortices at x = 0 is equal to half of the value (130) in the Trefftz
plane at x = −∞, or
s2
vT ( 0, 0, z ) =
d Γ dζ
1
−
.
4π − ∫s 2 dζ ζ − z
(140)
Here the subscript T is used to denote the velocity due to the trailing vortices. In general, for z < 21 s, the integral in (140) is negative and vT is a
downwash.
Before using (140) to correct the inflow velocity vector in the inner solution, other possible corrections should be considered, associated with the
bound and free vortices within the foil. Since the length of the free vortices
is equal to the chord length, which is small compared to the span, the free
vortices will have a negligible influence on the downwash compared to the
trailing vortices. The bound vortices are more difficult to treat, but from
the Biot-Savart integral (113) it can be shown that the correction due to the
spanwise variation of γB is negligible by comparison with (140).
Returning to the inner solution, the most significant effect of the threedimensional corrections is to impose a vertical velocity component given
by (140), or to change the effective angle of attack by the induced angle
s2
αi (z) ≡
1
vT
d Γ dζ
−
=
.
U 4πU − ∫s 2 dζ ζ − z
(141)
The total circulation on a two-dimensional flat plate, and the flat-plate
loading portion of an arbitrary cambered foil, is Γ = πUlα. Since (141) is negative on the foil, the total circulation of the lifting line must be decreased,
by comparison with its two-dimensional value at each section, by an increment −πUlαi. Substituting (141) gives an equation for the total circulation
in the form
s2
Γ (z) = Γ 2 D(z) +
1
d Γ (ζ ) dζ
l( z ) ∫−
,
4
dζ ζ − z
−s 2
−
1
1
s < z < s.
2
2
(142)
Here Γ2D(z) denotes the two-dimensional circulation of the corresponding
section, including the effects of camber and of the geometric angle of attack
Lifting Surfaces
213
α, and the last term in (142) represents the change in the circulation due to
the induced angle of attack.
Equation (142) is Prandtl’s lifting-line equation. It is an integrodifferential equation, since the derivative of the unknown Γ appears in the
integrand. In this respect, (142) differs from the integral equation (60) that
governs the two-dimensional vortex distribution, and the complication is
sufficient that (142) cannot be solved for general planforms without resorting to numerical approximations. Before discussing this matter further, we
note that additional end conditions must be imposed to make the solution
of (142) unique, as was the case in the two-dimensional situation where the
Kutta condition was imposed. The appropriate physical condition requires
the circulation to vanish at the tips of the foil, Γ ( ± 21 s ) = 0 .
Once again, the Fourier sine series (136) may be used. The induced angle
of attack (141) takes the form
αi = −
π
∞
1 ∞
cos nθ ′ dθ ′
sin nθ
nan ∫−
= − ∑ nan
,
∑
π n =1
θ
θ
−
cos
cos
sin θ
′
n =1
0
(143)
and the lifting-line equation (142) reduces to a system of linear equations
for the coefficients an,
∞
∑a
n
sin n θ =
n =1
∞
1
1
π 1
sin n θ
Γ 2 D  s cos θ  − l  s cos θ  ∑ nan
,




2Us
2
2s 2
sin θ
n=1
0 < θ < π.
(144)
For an elliptical spanwise loading, an = 0 for n ≥ 2 and the induced angle
(143) is constant along the span.
A specific geometry, where (144) is satisfied by the elliptical loading, is
an uncambered lifting surface of elliptical planform,
l( z ) = l0 sin θ ,
(145)
where l0 is the chord at mid span. The two-dimensional circulation is given
in this case by Γ2D = πUαl, and it follows from (l44) that
a1 =
παl0 
π l −1
1+ 0  .
2s 
2s 
(146)
For the elliptical planform, the aspect ratio is given by A = 4s/πl0, and thus
(146) may be replaced by
214
a1 =
Chapter 5
2α
.
A+2
(147)
The lift and drag coefficients (137–138) take the simple forms
CL =
2π Aα
,
A+2
(148)
CD =
C2
4π Aα 2
= L .
2
( A + 2)
πA
(149)
As the aspect ratio tends to infinity, the two-dimensional results are recovered, with zero induced drag and a lift coefficient equal to 2πα.
A systematic approach to lifting-line theory is given by Van Dyke (1975),
based on the method of matched asymptotic expansions. This leads to
Prandtl’s result (142) as well as to higher-order approximations. A notable
simplification of this approach is to replace the unknown derivative Γ′,
to the first approximation, by its two-dimensional value, giving explicit
results without the complications of an integral equation.
To illustrate this simplification, let us express the spanwise circulation
in the form
Γ ( z ) = Γ ( 1) ( z ) + Γ ( 2 ) ( z ) + Γ ( 3 ) ( z ) + ,
(150)
where Γ(1) is the first approximation, or the two-dimensional circulation
Γ2D, and successive terms represent the corresponding corrections at each
stage of the approximation. By assumption Γ(2) ≪ Γ(1), etc., with the understanding that these orders of magnitude are consequences of the underlying assumption A ≫ 1. Since the integral term in Prandtl’s lifting-line
equation (142) is proportional to the chord length l(z), this term will be
inversely proportional to the aspect ratio, or O(1/A). Thus explicit solutions
for the first two terms in (150) can be written as follows:
Γ (1) ( z ) = Γ 2 D ( z ),
(151)
s2
Γ (2 )( z ) =
1
d Γ (1) (ζ ) dζ
.
l( z ) ∫−
4
dζ ζ − z
−s 2
(152)
To proceed beyond this point on the basis of (142) would not be consistent
with the approximations already made in deriving (142).
For the elliptic planform, the lift coefficient derived from (151–152) is
readily obtained, in the form
Lifting Surfaces
215
CL(1) = 2πα ,
(153)
CL(1) + CL( 2 ) = 2πα (1 − 2 A).
(154)
At first glance the second approximation (154) differs from Prandtl’s result
(148), but if the denominator of (148) is expanded in a Laurent series, in
inverse powers of the aspect ratio, it follows that
CL = 2πα (1 − 2 A) + O(1 A2 ).
(155)
Thus we see that these results are consistent, up to the point where a comparison is justified.
These results are shown in figure 5.22, which includes the conventional
lifting-line lift coefficient (148) and the second-order systematic approximation (154), as well as a third-order approximation derived by Van Dyke
(1975) and exact lifting-surface calculations. The third-order approximation is not simply a more accurate Laurent series approximation of (148)
but a consistent theory including all terms1 of order 1/A2. One example of
Figure 5.22
Lift coefficient of elliptic wing. (From Van Dyke 1975, fig. 9.4)
216
Chapter 5
the corrections appearing at this order is an effective change not only in
the angle of attack but also in the curvature of the streamlines, which represents a change in the camber.
If one attempts to judge the merits of these different approaches from
figure 5.22, the conclusions are somewhat mixed. Prandtl’s “inconsistent”
lifting-line theory is remarkably useful for all aspect ratios from zero to
infinity. It overpredicts the lift coefficient, with a relative error of about 5
percent at A = 8, increasing to 10 percent at A = 4, 20 percent at A = 2, and
100 percent at A = 0. On the other hand, the consistent results are more
accurate for large aspect ratio but become completely invalid as the aspect
ratio is reduced. Thus, while Prandtl’s approximation is not systematic, the
most important higher-order effects seem to have been included through
good luck or shrewd insight or a combination of the two. This favorable
outcome is not observed often in asymptotic approximations, and contemporary researchers in marine hydrodynamics who lack Prandtl’s insight
should not expect to be so successful!
5.12 Cavity Flows
Cavitation may have a variety of causes. The most common example is
boiling water, where the vapor pressure is increased by raising the water
temperature. Generally in marine hydrodynamics applications, cavitation
results because the fluid pressure is reduced to the vapor pressure limit. This
will occur if the body velocity U is sufficiently large, since the pressure is
a decreasing function of fluid velocity. Alternatively, the ambient pressure
may be reduced by decreasing the body submergence and corresponding
hydrostatic pressure or by evacuating a variable-pressure water tunnel or
towing tank. Finally, cavitation may result if the body geometry is modified
to reduce the pressure, for instance, if the angle of attack of a hydrofoil is
increased.
The relevant nondimensional parameter describing the occurrence of
cavitation and the details of the resulting flow is the cavitation number
σ=
p0 − pv
.
1
ρU 2
2
(156)
Here p0 is the ambient pressure of the flow, usually at a large distance from
the body, and pv is the vapor pressure of the fluid.
Lifting Surfaces
217
Large cavitation numbers imply fully wetted flow, without cavitation.
As the cavitation number is reduced or the angle of attack increased, local
cavitation will occur first near the minimum-pressure position on the foil,
initially in the form of small isolated bubbles of vapor. For smaller values of
σ, the cavitation will become more widespread, and ultimately the flow will
become supercavitating, with the entire suction side of the foil contained
within the cavity, as shown in figure 5.23.
The lifting-surface theory outlined in the previous sections of this chapter can be used to predict the initial occurrence of cavitation in terms of
the minimum pressure on the foil. Subsequently, the cavitation bubbles
may become sufficiently large to affect the flow, generally in an unsteady
or unstable manner; in the ultimate supercavitating regime, however, the
cavity is relatively large and stable, and its effects can be accounted for by
a suitable generalization of lifting-surface theory. This procedure will be
illustrated in sections 5.13 and 5.14 for steady two-dimensional flow past
a symmetric wedge-shaped strut, as well as for the corresponding lifting
problem.
Figure 5.23
Supercavitating flow past a two-dimensional foil. Here the lower surface of the foil is
a circular arc, at 6° angle of attack, and the cavitation number σ = 0.19. (From Barker
and Ward 1976)
218
Chapter 5
As in the fully wetted case, we assume that the foil and the cavity are
thin. The boundary conditions are then linearized, as in section 5.2. On the
wetted portion of the body, the kinematic boundary conditions (11–12) are
unchanged. On the cavity, these are replaced by an appropriate dynamic
boundary condition prescribing the fluid pressure to be equal to the vapor
pressure. Using the linearized Bernoulli equation (14) and the cavitation
number (156), the appropriate dynamic boundary condition on the cavity
surface is given by
1
ρuU = pv − p0 = − ρU 2σ ,
2
(157)
or
1
u = − Uσ.
2
(158)
Strictly speaking, the hydrostatic pressure contribution to Bernoulli’s equation cannot be neglected in (157) and will give rise to a buoyancy force
acting on the cavity, but for sufficiently high speeds this effect is negligible
and will be ignored here. The survey articles by Wu (1968, 1972) and Acosta
(1973) discuss this and many other aspects of cavitation that are beyond
the scope of our present treatment.
5.13 Symmetric Cavity Flows
As our first example we consider the flow past a symmetric strut with a
blunt base, or trailing edge, as shown in figure 5.24. Here the coordinates
have been nondimensionalized in terms of the chord length of the strut,
with the leading edge at x = 1, and the nondimensional strut thickness is
2y0(x). Downstream of the trailing edge, at x = 0, a cavity of length l exists
with linearized pressure prescribed by (158). The resulting boundary conditions are shown in figure 5.25. The problem statement is completed by
requiring that Laplace’s equation must be satisfied in the fluid, the perturbation velocity and pressure should vanish at infinity, and the velocity at
the trailing edge of the foil is finite. The latter condition, like the Kutta
condition for the fully wetted lifting foil, is necessary to ensure a smooth
transition of the flow from the body surface to the cavity, with continuous
slope between these two boundaries.
Lifting Surfaces
219
Figure 5.24
Cavity flow past a symmetrical strut.
Figure 5.25
Linearized boundary conditions for cavity flow past a symmetric strut.
Since this problem is symmetrical about y = 0, it is logical to seek the
solution as a source distribution q(x), which represents the thickness effects
of both the strut and the cavity. From (54) the resulting velocity components on the boundary surfaces (−l < x < 1) are
1
u± ( x) = −
1 q(ξ )dξ
−
,
2π −∫1 ξ − x
1
v ± ( x) = ± q( x).
2
(159)
(160)
220
Chapter 5
From the boundary condition on the strut, it follows that
1
v ± = ∓Uy 0 ′ ( x) = ± q( x),
2
0 < x < 1.
(161)
This determines the source strength on the strut, in a manner identical to
the fully wetted thickness problem. Imposing the cavity boundary condition gives the integral equation
1
1
1 q(ξ )
−
− Uσ = −
dξ ,
2
2π −∫1 ξ − x
− l < x < 0,
(162)
or using (161),
1
0
1
1
1 q(ξ )
y ′ (ξ )
−∫
− Uσ − U ∫ 0
dξ = −
dξ ,. − l < x < 0
π 0 ξ−x
2
2π − l ξ − x
(163)
This is essentially the same integral equation as (60), which relates the vortex distribution γ of the fully wetted foil to its normal velocity v. The only
difference is that the interval (−1, 1) must be replaced by (−l, 0). The appropriate solution satisfying the Kutta condition at x = 0 is obtained from (73),
in the form
0
q( x) =
2  −x  1 2  l + ξ 
−
π  l + x  −∫l  −ξ 
12
1
 1
1
y ′ (t )  dξ
,
dt 
× − U σ − U ∫ 0
π 0 t − ξ  ξ − x
 2
(164)
− l < x < 0.
Here the dummy variable t has been substituted for x in the inner integral.
The integrals with respect to ξ in (164) can be evaluated using the Hilbert
transforms derived in section 5.7. The treatment of the first term in braces
is straightforward, using (81) and (83); in the second term, however, the
order of integration must be inverted, the denominator expanded in partial
fractions, and (88) used with (82). The final result is
−x  1 2  2  −x  1 2
q( x ) = −U σ 
− U
 l + x
π  l + x
1
12
l + t  y 0′ (t ) 
− l < x < 0.
dt ,
× ∫
 t  t − x 
0
(165)
Substitution of the source strength (165) in (159) and (160) gives the
velocity and pressure distribution on the strut and the cavity surface. The
Lifting Surfaces
221
cavity shape may be found by imposing the kinematic boundary condition
(161) on the cavity surface and integration of (165). In particular, requiring
the cavity to be closed implies that the integral of the source strength, over
the body plus the cavity, must be equal to zero. Integrating (161) and (165)
gives the relation
1
1
1

 l + t 1 2
π
0 = ∫ q( x)dx = − U σ l − 2U ∫ 
− 1 y 0′ (t ) dt − 2U ∫ y 0′ (t )dt .
 t 
2


0
−l
0
(166)
Here the last integral comes from the contribution of the sources in the
segment (0, 1), and (82), (88) have been used. Reduction of (166) gives an
equation between the cavitation number σ and cavity length l,
1
0=
π
l + t 1 2
σl + ∫ 
y 0′ (t ) dt .
 t 
4
0
(167)
Since the pressure acting on the wetted surface of the strut is greater than
the vapor pressure pv, acting on the base, a positive drag force will result,
equal to
1
D = −2 ∫ ( p − pv ) y 0′ ( x) dx
0
1
1
= −2 ρ ∫  uU + U 2σ  y 0′ ( x ) dx.


2
0
(168)
Using equation (159) for the velocity u gives
D=
1 1

ρU  q(ξ )
dξ − πU σ  y 0′ ( x)dx.
−
∫
π 0  −∫l ξ − x

(169)
By substituting equations (161) and (165) for the source strength q(x) on
the body and cavity respectively, and (167) for the cavitation number, and
by using the Hilbert transforms (80–88), we may express (169) in the form
1
D=
1
12
12
ρU 2
 x  × − y ′ (t )  l + t   4 − 2  dt dx.
′
y
(
x
)
0
0
∫
 t   l t − x
 l + x
π ∫0
0
(170)
Equation (170) can be recast in a simpler form as follows. First we invert the
order of integration. (The more rigorous reader may find justification for
this operation in Muskhelishvili 1953, p. 59.) If the dummy variables (x, t)
are inverted at the same time, an alternative expression is obtained:
222
Chapter 5
1
D=
1
ρU 2
l + x1 2
t 1 2  4
2 
y0′ ( x) 
dt dx.
× ∫− y 0 ′ (t ) 
+
∫
 x 
 l + t   l l − x
π 0
0
(171)
Since (170) and (171) are equivalent, the drag can be written as one half of
their sum,
D=
1
1
 2  x(l + t )  1 2  t (l + x)  1 2 
ρU 2
′
′
−
y
(
x
)
y
(
t
)
0
0
 
 + 
 
∫
π ∫0
x(l + t )  
 l  t (l + x) 
0
12
12
1  t (l + x ) 
 x(l + t )   
+

 − 
   dt dx.
t − x  x(l + t )
t (l + x )
 
(172)
After algebraic reduction of the integrand it follows that
1
D=
1
ρU 2
(l + 2 x )(l + 2t )
y 0 ′ ( x)∫ y ′ 0 (t )
dt dx,
π l ∫0
[
(l + x )(l + t )]1 2
xt
0
(173)
and thus
2
D=
1

ρU 2 
(l + 2 x )
y0′ ( x)
dx .
∫
12

πl  0
[ x(l + x)]

(174)
The integrals in equations (167) and (174) can be evaluated without difficulty for simple strut shapes. Figures 5.26 and 5.27 show the values of the
cavity length and drag force for a simple wedge, as well as two parabolic
struts. In all cases the chord length is unity. These figures show that for
decreasing values of the cavitation number σ, the cavity length increases
while the drag force is reduced. Ultimately, l → ∞ as σ → 0, whereas the drag
force tends to a finite limiting value dependent upon the body shape. A
comparison of the results for these three strut geometries demonstrates the
importance of the trailing-edge slope y 0′ ( 0 ) on the subsequent development
of the cavity. In particular, a transition from the convex parabola, where
y 0′ ( 0 ) = 0, to the wedge and ultimately the cusped case, results in increased
cavity length and drag.
The linear theory becomes invalid if the cavity length is substantially
shorter than the chord length, or if the cavitation number is too large. This
restriction is apparent from the linearized boundary condition (158) on the
cavity surface, which implies that u/U ≪ 1 only if σ ≪ 1. Further discussion
of this matter and a comparison of the nonlinear theory with experiments
may be found in the paper by Wu, Whitney, and Brennen (1971).
Lifting Surfaces
223
Figure 5.26
Cavity length l for the three symmetric struts shown. Note that y0(0) and l are nondimensionalized with respect to the unit chord length.
5.14 Supercavitating Lifting Foils
To simplify the problem of a lifting foil in the supercavitating flow regime
as much as possible, we will assume that the cavitation number is zero and
that the foil geometry is as sketched in figure 5.28. In particular, the leading edge is sharp, and the entire upper side of the foil is situated within the
cavity.
Nondimensional coordinates (x, y) are chosen such that the chord
length is equal to unity, and the slope of the lower surface of the foil is
denoted by the local angle of attack α(x). For reasons of subsequent
224
Chapter 5
Figure 5.27
Drag coefficient for the three symmetric struts shown in figure 5.26, nondimensionalized in terms of the unit chord length.
Figure 5.28
Supercavitating flow past a lifting foil with sharp leading edge.
Lifting Surfaces
225
Figure 5.29
Linearized boundary conditions corresponding to the problem shown in figure 5.28.
Figure 5.30
Boundary conditions in mapped ζ-plane.
algebraic convenience, the leading edge is situated at the origin; the linearized boundary conditions on the foil and cavity then take the form shown
in figure 5.29.
This problem cannot be decomposed readily into symmetric and antisymmetric portions, as in the fully wetted problem, because of the fundamentally different boundary conditions imposed on the upper and lower
sides of the segment occupied by the foil in figure 5.29. However, the conformal mapping function ζ = −iz1/2 can be used to map the domain of the
fluid into the lower half of the ζ-plane, folding the z-plane about the leading edge of the foil. The complex velocity u − iv is an analytic function of
z, and hence also of ζ, which may be mapped from one plane to the other
with its values preserved.
The resulting problem in the ζ-plane is as shown in figure 5.30, with ζ
= ξ + iη. In this mapped plane, the vertical velocity on the lower surface of
the foil has been written as
226
v = −U α ( x) = −U α ( −ξ 2 ) ≡ f (ξ ).
Chapter 5
(175)
Since the fluid domain has been mapped entirely into the lower half of
the ζ-plane, the region η > 0 has no physical significance. Mathematically,
it corresponds to the second Riemann sheet of the z-plane, or it could be
regarded as the domain of the cavity. In any event, if the boundary conditions are satisfied as shown in figure 5.30, on the lower side of the real
axis, the flow in the upper half-plane is immaterial. In accordance with the
Schwarz reflection principle, the velocity components u and v can therefore
be reflected about the ξ-axis, and the resulting flow will be continuous across
the real axis, excluding the cut (−1, 0), provided u is an odd function of η.
To ensure that u − iv is analytic, it follows from the Cauchy-Riemann equations that v must be an even function of η. Thus, the appropriate reflections
into the upper half-plane are given by
u(ξ , η ) = − u(ξ , −η ),
(176)
v(ξ , η ) = v(ξ , −η ).
(177)
The reflected boundary-value problem shown in figure 5.31 is mathematically identical to the lifting problem of a fully wetted thin foil shown
in figure 5.14. In both cases a Kutta condition must be imposed at the trailing edge, and the perturbation velocity should vanish at infinity. Thus the
solution of the cavity-flow problem in the mapped plane is identical to that of the
fully wetted lifting problem in the physical plane.
The solution is given by a line distribution of vortices along the cut (−1,
0), as in equation (55); the appropriate distribution satisfying the Kutta
condition at x = −1 can be inferred from (71),
Figure 5.31
Boundary conditions of the reflected complex velocity in the ζ-plane.
Lifting Surfaces
1
2  1+ ξ
γ (ξ ) = 
π  −ξ 
2
227
12 0
12
 −t  f (t ) dt ,
t −ξ
−1
∫−  1 + t 
− 1 < ξ < 0.
(178)
On the lower surface of the foil, the horizontal velocity component u_(ξ)
is equal to
1
2
γ (ξ ), as in (59), and the lift force can be derived from the
linearized Bernoulli equation,
L=
0
0
−1
−1
∫ ( p − pv )dx = ρU ∫ u− ( x) dx.
(179)
To evaluate the last integral, the variable of integration may be changed
from the physical coordinate x to the mapped coordinate ξ. Thus,
0
L=
0
1
dx
ρU ∫ γ (ξ ) dξ = − ρU ∫ γ (ξ )ξ dξ.
ξ
2
d
−1
−1
(180)
This can be recognized as the negative of the moment, about the leading
edge, of a fully wetted foil of unit chord length. Thus, with α(x) the slope
of the supercavitating foil and η(x) the mean-camber line of an equivalent
fully wetted foil,
α ( − x2 )cavitating foil = ( dη dx )equivalent fully wetted foil .
(181)
The supercavitating lift coefficient is given by
(CL )cavitating = −(CM )equivalent .
(182)
Similarly, the moment coefficient for the cavitating foil is equal to the third
moment of the equivalent fully wetted foil.
The drag coefficient of the cavitating foil can be related to the lift coefficient of the fully wetted foil, following an analysis similar to that used in
deriving (174). The drag force is obtained from (179) after multiplying the
integrand by the slope α(x):
0
0
−1
−1
D = ρU ∫ u− ( x)α ( x)dx = − ρ ∫ u− ( x)v( x)dx.
(183)
Here the boundary condition (I75) has been used to replace the slope by
the vertical velocity component. Changing the variable of integration, as
in (180), gives
228
Chapter 5
0
D = ρ ∫ γ (ξ )v(ξ )ξ dξ.
(184)
−1
Using (60) for the vertical velocity gives
D=−
ρ
2π
0
0
γ (t )
dt dξ.
t −ξ
−1
∫ γ (ξ )ξ ∫−
−1
(185)
If the order of integration and dummy variables are inverted, as in (171), it
follows that
D=−
=
ρ
4π
ρ
4π
0
0
t 
 −ξ
∫ γ (ξ ) ∫− γ (t )  t − ξ + t − ξ  dt dξ
−1
−1
0
0
−1
−1
∫ γ (ξ ) ∫ γ (t )ddt dξ
(186)
2
=
0

ρ 
γ (ξ ) d ξ  .
4π  −∫1

The last integral is the total circulation Γ for the equivalent foil, and thus
the drag coefficient for the supercavitating foil is given by
(CD )cavitating =
1
(CL 2 )equivalent .
8π
(187)
This useful equivalence was first established by Tulin and Burkart in
1955 and has been expounded in several papers (Tulin 1964). For a flat
plate of constant angle of attack α, the results from section 5.4 give for the
corresponding supercavitating case
CL =
π
α,
2
(188)
CM =
5πα
,
32
(189)
CD =
π 2
α .
2
(190)
Here the center of pressure is shifted from the quarter-chord point to a
position 5/16 of the chord downstream from the leading edge. Of greater
interest is the lift-drag ratio, which reveals that the total force vector on
the foil is precisely normal to the flat plate, inclined at an angle a from the
Lifting Surfaces
229
vertical. This situation differs from the fully wetted foil, where the force
vector is strictly vertical, due to the leading-edge suction force. In the supercavitating case there is no leading-edge suction force, since the square-root
singularity of the equivalent fully wetted flow in the ζ-plane is reduced to
a quarter-root singularity in the physical z-plane. Physically, the cavity that
originates at the leading edge is of more gradual curvature than the flow
around the fully wetted flat plate.
More generally, the lift-drag ratio can be derived from (182) and (187) as
( L / D )cavitating = 8π (CM / CL 2 )equivalent .
(191)
To maximize this quantity, the ratio CM/CL for the equivalent foil must be
as large as possible, or the center of pressure must be as far downstream as is
practical. Since the pressure on the lower surface must be positive to avoid
face cavitation, the optimum lift-drag ratio results when the equivalent
center of pressure is at the trailing edge. This situation cannot be attained in
practice, but it does imply correctly that efficient supercavitating sections
will have a pronounced maximum slope at the trailing edge.
While the equivalence of supercavitating and fully wetted foils is important in predicting the performance and optimizing the design of supercavitating foils, the performance of these two types of foils is not identical. As a
simple example, the flat-plate lift coefficient is reduced by a factor of four
relative to the fully wetted regime, and a nonzero drag force is incurred as
well. Thus the supercavitating flow regime is to be avoided if possible, especially given the problems of erosion and noise, which are inevitable consequences of cavitating flows. However, if operation in the supercavitating
regime is inevitable, it is preferable to design the foil section to be optimum
in this regime, and thus to have a very different form from the fully wetted
foil shapes. The latter difference is particularly important at the leading
edge where, to avoid both cavitation and separation, a fully wetted foil is
designed with a substantial radius of curvature, whereas a supercavitating
foil is designed with a sharp leading edge.
5.15 Unsteady Hydrofoil Theory
In general, unsteady motions will be accompanied by corresponding changes in the circulation about the foil or about each section of a
230
Chapter 5
three-dimensional lifting surface. We shall confine our attention here to
the two-dimensional case, since the situation in three dimensions is qualitatively similar except for the addition of suitable trailing vortices. Applying Kelvin’s theorem to a large contour surrounding the foil and its wake
during the entire history of its motion, as in figure 5.6, we find that the
circulation about this contour must be zero. Thus any change in the circulation Γ(t) about the foil must be balanced by an equal and opposite change
in the vorticity shed into the wake. The starting vortex shown in figure
5.6 is a particular example where the unsteady effects are restricted to the
initial acceleration of the foil, but in the more general case vorticity will be
continuously supplied into the wake.
The existence of a vortex distribution throughout the wake is reminiscent of the three-dimensional steady lifting problem of section 5.9. As in
that case, the vortices in the wake are responsible both for a drag force acting on the foil and for a downwash which affects the lift. However, unlike
the steady lifting-surface problem with its trailing vortices, the present
case is purely two-dimensional, and the vortices in the wake must be oriented in the same manner as the bound vorticity on the foil, as shown in
figure 5.32.
The downwash induced at the foil by the wake is time-dependent; each
individual vortex element induces a contribution to the downwash which
diminishes with time as the vortex moves downstream. In this respect, a
memory is associated with the motion of the foil: the lift force at a given
moment will depend on the integrated downwash from the entire wake and
thus on the previous time-history of the motion. This memory effect is one
of the principal features of the unsteady hydrofoil problem. An analogous
situation will be encountered in chapter 6, where the unsteady motions of
Figure 5.32
Unsteady vortex wake behind an oscillating foil. The case shown is for heaving motion of the foil, with a reduced frequency k = 1.0. The orientation and length of each
arrow denotes the sign and strength of the vorticity at the corresponding position
in the wake.
Lifting Surfaces
231
a floating body will generate waves which, in turn, affect the body over a
subsequent period of time.
Our discussion of unsteady hydrofoil theory will be limited to the simplest two-dimensional case—linearized flow with constant forward velocity. The linear restriction implies not only that the foil is thin and at a small
angle of attack, but that the unsteady motions are small. As in the steady
case, the foil can be collapsed onto a cut in the x-axis, and similarly the vortex sheet in the wake is confined to the x-axis. Elsewhere the fluid motion
is irrotational, with the perturbation velocity field defined by a velocity
potential that vanishes at infinity. Other conditions to be imposed include
a kinematic boundary condition on the foil, a Kutta condition at the trailing edge, and a dynamic boundary condition on the vortex wake. Finally,
on the assumption that the motion has arisen from an initial state of rest,
the total circulation about the foil plus its wake must vanish.
As in sections 5.6 and 5.7, it will be convenient to assume that the coordinates are nondimensionalized on the basis of the half-chord 21 l, with the
leading and trailing edges of the foil at x = ±1, respectively, and the wake
confined to the line x < −1.
Thickness and lifting effects can be treated separately because of linearization. Unless the thickness is variable in time the unsteady effects will
be confined to the lifting problem. Thus we consider an unsteady meancamber line, or a foil of zero thickness with camber and angle of attack that
vary with time in some prescribed manner.
If the vertical coordinate of the mean-camber line is denoted by y = η(x,
t), the kinematic boundary condition on the foil takes the form
0=
∂
∂
D
+ ∇φ ⋅ ∇  ( y − η )
( y − η) =  − U
 ∂t

∂x
Dt
∂φ ∂η
∂η
,
=
−
+U
∂y ∂t
∂x
(192)
where second-order terms in ϕ and η are neglected. Applying this condition on the cut gives an equation for the vertical velocity component on
y = 0,
v ( x, 0 , t ) =
∂η
∂η
≡ v 0 ( x, t ),
−U
∂t
∂x
− 1 < x < 1.
(193)
This is the appropriate generalization of the boundary condition (22) for
the steady-state lifting problem.
232
Chapter 5
While various techniques can be used to solve this problem, it is natural
to seek a generalization of the corresponding steady-state case and thus to
represent the foil and its wake by a distribution of vortices. With an obvious
extension of (59–60), the velocity components on the upper or lower sides
of this vortex sheet will be
1
u± ( x, t ) = ∓ γ ( x, t ),
2
(194)
and
v ± ( x, t ) = −
1
2π
1
∫
−∞
γ (ξ , t )
dξ.
ξ−x
(195)
The condition that the total circulation about the foil plus its wake should
equal zero requires that
1
−1
−∞
−∞
∫ γ ( x,t ) dx =
∫ γ ( x,t )dx + Γ (t ) = 0.
(196)
The linearized pressure is obtained as in (14), but includes the unsteady
contribution from (4.26). Thus
∂φ
∂φ
p − p∞ = − ρ 
−U .
 ∂t
∂x 
(197)
The dynamic boundary condition on the wake requires that the jump in
(197) across the wake should be zero. Differentiating (197) with respect to x
and substituting (194), we get
∂γ
∂γ
−U
= 0,
∂t
∂x
− ∞ < x < −1.
(198)
This is a simple partial differential equation for γ whose general solution is
γ ( x, t ) = γ ( x + Ut ),
− ∞ < x < −1.
(199)
Thus, as expected from Kelvin’s theorem, the vorticity is convected downstream with velocity U and remains constant in a reference frame moving
with the fluid.
If (196) is differentiated with respect to time and (198) utilized to integrate over the wake, it follows that
Lifting Surfaces
233
dΓ
= −U γ ( −1, t ).
dt
(200)
This equation relates the vorticity shed at the trailing edge to the rate of
change of Γ, in precisely the sense that was anticipated initially.
With the vortices in the wake described by (199), we can now construct
the complete solution, including an arbitrary vortex distribution on the
foil. Separating the integrals over the foil and wake and imposing the kinematic boundary condition (193) on the vertical velocity (195) gives
1
2π
v 0 ( x, t ) +
−1
1
γ (ξ + Ut )
1 γ (ξ , t )
∫ ξ − x dξ = − 2π ∫− ξ − x dξ,
−∞
−1
−1 < x < 1.
(201)
This is a singular integral equation for the vortex strength; it is identical to
(60) except that the integral2 on the left side of (201) is unknown. Ignoring
this temporarily, the general solution is given by (73–76) in the form
γ ( x, t ) =
 1 (1 − ξ 2 )1 2
[v 0 (ξ , t )
 ∫−
 −1 ξ − x
−1

1 γ (ξ ′ + Ut )
1 
+
∫ ξ′ − ξ dξ′  dξ + 2 Γ  .
2π −∞
2
π (1 − x2 )1 2
(202)
The double integral in (202) can be reduced in a manner similar to
(164), by interchanging the order of integration, using partial fractions, and
(81–88). Thus it follows that
γ ( x, t ) =
 1 (1 − ξ 2 )1 2
2
v 0 (ξ , t ) d ξ
−
π (1 − x2 )1 2  −∫1 ξ − x
−1

1 (ξ 2 − 1)1 2
γ (ξ + Ut ) dξ  .
+ ∫
2 −∞ ξ − x

(203)
Here (196) has been used.
If the Kutta condition is invoked as in (77), the terms in braces in (203)
must vanish at x = −1; this results in an integral equation for the wake
vorticity,
−1
 ξ − 1
∫  ξ + 1
−∞
12
1
 1− ξ
γ (ξ + Ut )dξ = 2 ∫ v 0 (ξ , t ) 
 1 + ξ 
−1
12
dξ.
(204)
If the wake consists of a single starting vortex far downstream with circulation −Γ, (204) reduces to the steady-state result (34). In general, Laplace
234
Chapter 5
transforms can be used to solve the integral equation (204) for the vortex
density in the wake, in a manner outlined by Woods (1961).
Before attempting to deal with (204), we shall compute the lift force and
moment acting on the foil. Using Bernoulli’s equation (197), the lift force
is given by
∂φ
∂φ +
L = ρ∫ 
− U  dx.
 ∂t
∂x  −
−1
1
(205)
From the symmetry of the lifting problem, the velocity potential is odd
in y and thus must vanish on the real axis upstream of the leading edge.
There is a square-root infinity in the velocity components at x = 1, implying a square-root zero of the velocity potential at the leading edge. Thus it
follows that
1
1
∂φ
1
∂φ
∂u
∫ ∂t dx = ∫ ∂t d(1 + x) = − ∫ (1 + x) ∂t dx.
−1
−1
(206)
−1
If this result with (194) is substituted in (205), the lift is given by
1
∂γ
L = ρ ∫ (1 + x )
+ U γ  dx.
t
∂


−1
(207)
Similarly, the moment about x = 0 is obtained in the form
∂φ
∂φ +
M = ρ∫ 
− U  x dx
 ∂t
∂x  −
−1
1
1
1
∂γ
= ρ ∫  − (1 − x2 )
+ U γ x  dx.
∂t
 2

−1
(208)
Substituting the vortex distribution (203), interchanging orders of integration, and using (81–88) in the usual manner gives
1
L = −2 ρ ∫ (1 − ξ 2 )1 2
−1
∂v 0 ( ξ , t )
dξ + ρU Γ
∂t
−1
∂γ (ξ + Ut )
dξ.
− ρ ∫ [1 + ξ + (ξ − 1) ]
∂t
−∞
2
(209)
12
The wake integral in (209) can be integrated by parts after substituting (198)
for ∂γ/∂t. Using (196), the expression for the lift force reduces to the form
Lifting Surfaces
235
1
L = −2 ρ ∫ (1 − ξ 2 )1 2
−1
−1
ξ dξ
∂v 0 ( ξ , t )
.
dξ + ρU ∫ γ (ξ + Ut ) 2
(
ξ
∂t
− 1)1 2
−∞
(210)
By similar analysis starting with (208), the moment about x = 0 is given by
1
M = − ρ ∫ ξ (1 − ξ 2 )1 2
−1
∂v 0 ( ξ , t )
dξ
∂t
1
= −2 ρU ∫ (1 − ξ 2 )1 2 v 0 (ξ , t )dξ
(211)
−1
=+
−1
dξ
1
ρU ∫ γ (ξ + Ut ) 2
.
− 1)1 2
2
(
ξ
−∞
It may be confirmed that (210) and (211) reduce to equations (15) and (37),
respectively, if the motion is independent of time and the wake consists
only of a starting vortex of circulation −Γ at x = −∞.
The first term for the lift in (210) is the added-mass force associated
with a generalized vertical acceleration along the plate. Likewise, the first
term in (211) is the added moment of inertia, and the second term gives
the moment resulting from equation (4.99). The last integrals in (210) and
(211) represent the effect of the wake; these introduce memory effects that
depend on the past history of the motion.
It is curious that if the moment is transformed to the quarter-chord position, the memory effects associated with the wake vanish. To show this, we
use (210) and (211) to compute the moment about the quarter-chord point
in the form
1
M−
1
L ≡ M x = 1 = −2 ρU ∫ (1 − ξ 2 )1 2 v 0 (ξ , t ) dξ
2
2
−1
1
+ ρ ∫ (1 − ξ )(1 − ξ 2 )1 2
−1
+
−1
1
 ξ − 1
ρU ∫ 
 ξ + 1
2
−∞
∂v 0 (ξ ,tt )
dξ
∂t
(212)
12
γ (ξ + Ut )dξ.
Using (204) to replace the last term, we get
1
M x= 1 = ρ ∫ (1 − ξ )(1 − ξ 2 )1 2
2
−1
1
∂v 0 ( ξ , t )
dξ
∂t
1− ξ
− ρU ∫ (1 + 2ξ ) 
 1 + ξ 
−1
12
v 0 (ξ , t ) dξ.
(213)
236
Chapter 5
From (213) it is clear that the moment about the quarter-chord point is
not affected by memory-dependent downwash from the wake and can be
expressed in terms of the instantaneous normal velocity and acceleration
of the plate. This is not to say that the moment is independent of the vortices in the wake, which give a contribution represented by the last term
of (212). Since (213) can be calculated directly for any prescribed motion
of the foil, it is only necessary to consider the lift force, which can be combined with (213) to give the moment about any point along the foil.
5.16 Oscillatory Time Dependence
The simplest case of unsteady motion is sinusoidal time dependence, with
period 2π/ω. In this case the normal velocity of the foil can be expressed in
the form
v 0 ( x, t ) = Re[ v 0 ( x)e iωt ],
(214)
and the wake vorticity must be of a similar form,
γ ( x + Ut ) = Re[γ 0 e i(ωt + kx ) ].
(215)
Here Re denotes the real part, and k = ω/U is the reduced frequency. This
parameter can be interpreted physically as π times the ratio of the chord
length to the wavelength of the wake vortices.
Substitution of (214–215) in (203) gives an equation for the unknown
constant γ0 in the form
1
1− ξ
γ 0 = 2∫ 
 1 + ξ 
−1
12
v 0 (ξ ) d ξ
−1
 ξ − 1
∫  ξ + 1
12
e ikξ dξ .
(216)
−∞
Postponing consideration of the numerator, which will depend on the prescribed motion of the foil, we consider the integral in the denominator
which depends only on the reduced frequency k. Strictly, this integral is not
convergent because the oscillatory motion assumed in (214–215) violates
the assumption of an initial state of rest. One technique for avoiding this
difficulty is to make ω and k slightly complex, with negative imaginary
parts, so that (214) and (215) vanish for t → −∞. An equivalent but simpler
approach is to assume that the motion starts abruptly from a state of rest
at some time t0 ≪ t. The vorticity (215) is replaced by zero if x < (t0 − t)U,
Lifting Surfaces
237
and the infinite lower limit of the integral in the denominator of (216) is
replaced by (t0 − t)U.
With the latter interpretation, the denominator of (216) becomes
−1
 ξ − 1
 ξ + 1
( t0 −t )U
∫
12
−1
dξ
e ikξ .
2
(
− 1)1 2
ξ
( t0 −t )U
e ikξ dξ = (1 + i d dk )
∫
(217)
In the limit t0 → −∞, the integral on the right-hand side of (217) is proportional to the Hankel function H0(2)(k), where Hn(2)(k) = Jn(k) − iYn(k) defines
the Hankel function in terms of the Bessel functions of the first and second
kinds. The necessary properties of these functions are given by Abramowitz
and Stegun (1964). Thus it follows that
−1
 ξ − 1
 ξ + 1
t0 →∞
( t0 −t )U
lim
∫
12
e ikξ dξ = −
π
i(1 + i d dk )H 0( 2 ) ( k )
2
π
= − [ iH 0( 2 ) ( k ) + H1( 2 ) ( k )].
2
(218)
If this result is substituted in (216), the parameter γ0 is determined as
γ0 = −
1
4 1− ξ
π −∫1  1 + ξ 
12
v 0 (ξ ) dξ {iH 0( 2 ) ( k ) + H1( 2 ) ( k )}.
(219)
The wake integral in (210) can be evaluated in a similar manner, with (215)
and (219) used for γ(ξ + Ut). The lift force is given by the expression
1
H (2 )(k )

  1− ξ
L = −2 ρU e iωt  ( 2 ) 1
 ∫

(2)

 H1 ( k ) + iH 0 ( k )  −1  1 + ξ 
1

+ ik ∫ (1 − ξ ) v 0 (ξ ) dξ  .

−1
12
v 0 (ξ ) d ξ
(220)
2 12
Thus, for sinusoidal time dependence, the memory effects of the wake
can be expressed as frequency-dependent force coefficients proportional to
the ratio of the Hankel functions:
C( k ) =
H
(2)
1
H1( 2 ) ( k )
.
( k ) + iH 0( 2 ) ( k )
(221)
This ratio is called the Theodorsen function and is plotted in an Argand
diagram in figure 5.33. For low frequencies, the limiting value of the
238
Chapter 5
Figure 5.33
The Theodorsen function (221) representing the reduction in amplitude and shift in
phase of the lift force due to a vertical oscillation h. (From Bisplinghoff, Ashley, and
Halfman 1965)
Theodorsen function is 1.0, and the lift and moment reduce to their quasisteady-state values.
The cases of greatest practical interest are oscillatory rigid-body motions
in the vertical direction (heave) and about the origin (pitch). These displacements of the foil will be denoted by
h(t ) = Re h0 e iωt ,
(222)
and
α (t ) = Re α 0 e iωt ,
(223)
respectively, where h0 and α0 are complex constants. The resulting oscillatory mean-camber line is η = h + αx, and the normal velocity follows from
(193),
v 0 ( x, t ) = h + α x − U α .
(224)
Here a dot denotes the time derivative. Substituting these results with (214)
in (220) and evaluating the integrals using (78–79), we find that the lift
force is given by
{
}
1
L = −2πρU 2 Re e iωt C( k ) ikh0 −  1 + ik  α 0  − πρ( h − U α ).

2  

(225)
The last term in (225) has been left in a general time-dependent form to
emphasize its origin in the added-mass sense. This contribution to the lift
force can be derived from equation (4.115), after noting that the velocity of
the foil in a body-fixed reference frame is given by
Lifting Surfaces
(U cos α + h sin α , h cos α − U sin α ) (U , h − U α ).
239
(226)
No consideration has been given to the drag force, which results in part
from the normal component on the foil and in part from the leading-edge
suction force. This component is particularly important for flapping propulsion and has been studied extensively in the context of bird and fish
propulsion. A particularly simple result follows by considering the heaving
motion of a flat plate; here there is no normal component of the drag force
along the plate, and the leading-edge suction force is by its nature a negative drag, or positive propulsion force. This interesting area of unsteady
hydrofoil theory has been studied by Wu (1971) and by other authors cited
in that reference. More recent contributions can be found in Wu, Brokaw,
and Brennen (1975).
5.17 The Sinusoidal Gust Problem
If the velocity of the hydrofoil is steady, but the surrounding fluid is in a
state of nonuniform motion, the resulting interaction between the foil and
the fluid will be unsteady. Examples of this are the motion of hydrofoils in
waves and of propeller blades in a spatially nonuniform wake. This type of
unsteady lifting-surface problem, which is known as a gust problem, differs from those problems treated in sections 5.15 and 5.16 where the foil
is unsteady and the undisturbed velocity field of the surrounding fluid is
uniform.
The boundary condition on the foil, in the presence of a gust, can be
derived from (192–193) by adding the gust potential ϕg to the perturbation
potential ϕ associated with the foil. The boundary condition (193) takes the
modified form
v ( x, 0 , t ) =
∂n
∂n
−U
− v g ( x, 0, t ),
∂t
∂x
(227)
where vg denotes the upwash of the gust on the foil. Thus, if the foil is fixed,
the results obtained in sections 5.15 and 5.16. can be utilized for the gust
problem simply by setting v0 = −vg.
We shall restrict our attention here to gusts that are sinusoidal functions
of x + Ut, and thus steady with respect to the free stream. If α0 is the amplitude of the resulting change in angle of attack, vg can be written in the form
240
v g ( x, 0, t ) = Re{U α 0 e ikx + iωt }.
Chapter 5
(228)
Substituting this quantity for −v0 in (220) and evaluating the resulting integrals in terms of the Bessel functions J0 and J1 gives the lift force in the form
2i / π k



L = −2 ρU 2 Re α 0 e iωt  ( 2 )
(2)
.
+
H
(
k
)
iH
(
k
)
 1


0
(229)
The quantity in brackets in (229) is the Sears function, which is plotted in
figure 5.34. For large argument k, the Hankel functions in the denominator
oscillate proportionally to e−ik, and thus the phase of (229) varies rapidly, as
indicated in figure 5.34. This can be avoided if the phase of the sinusoidal
gust is measured with reference to the leading edge rather than the midchord position, and thus (229) is multiplied by the quantity e−ik. The result
is slowly varying, as shown in figure 5.34. Physically, this change suggests
that the foil responds to the gust most significantly when the gust is at the
leading edge. For this reason the leading edges of propeller blades are often
raked, so that the phase of the unsteady load along the blade will be distributed more uniformly as the blade enters a nonuniform region.
Figure 5.34
The Sears function for a sinusoidal gust, corresponding to the factor in brackets in
(229). The phase of the gust is measured relative to mid chord or leading edge in the
respective cases shown. Figures marked on the curves are the values of the reduced
frequency k.
Lifting Surfaces
241
5.18 Transient Problems
As noted in section 5.15, the general case of arbitrary time-dependence can
be treated directly by Laplace transform techniques. Equivalently, the frequency ω in the oscillatory time-dependent results of section 5.16 can be
made imaginary, and the superposition of these solutions used to generate
an arbitrary transient disturbance. We shall avoid performing this analysis
here, but two simple results will be noted—where the vertical velocity of
the foil is changed suddenly in a step-function manner, and where the foil
moves into a similar gust.
Since any constant angle of attack may be added without affecting the
unsteady problem, we assume that initially the foil moves with zero angle
of attack, and at time t = 0 this is changed by imposing a small vertical
velocity component. The resulting lift is given proportional to Wagner function shown in figure 5.35. This lift force immediately achieves a nonzero
value precisely half of the steady-state value and subsequently increases
monotonically to that limit. To develop 90 percent of the steady lift, the foil
must travel about six chord lengths.
This problem is identical to that of a foil accelerating instantaneously
from rest at t = 0 to a constant velocity U, with a constant angle of attack,
since in both cases the perturbation vanishes for t < 0. Thus the Wagner
function in figure 5.35 predicts the initial development of lift during the
rapid acceleration of a foil with constant α.
An inverse situation will develop if the foil stops suddenly. This case can
be related to the Wagner function by subtracting a constant angle of attack
equal to the value before stopping. On this basis it follows that the lift
force upon stopping is reduced immediately to half of its steady value, and
subsequently a more gradual attenuation to zero persists over a relatively
long period.
The corresponding gust problem is that of a sharp-edged gust. The resulting lift force, known as the Kussner function, is shown in figure 5.35. In this
case the lift force rises from an initial value of zero. Half of the ultimate
steady-state lift develops before the gust passes the trailing edge, and 90
percent is developed when the foil moves a distance of about eight chord
lengths beyond the gust. The details of this and other transient motions are
given by Woods (1961).
242
Chapter 5
Figure 5.35
Transient lift force due to a sudden change in vertical velocity (Wagner’s function)
and due to passage through a sharp-edged gust (Kussner’s function). The ordinate
denotes the number of chord lengths traveled after increasing the angle of attack, or
after encountering the gust at the leading edge.
Problems
1. Integrate the normal and tangential components of the complex velocity
(51) around a circular contour centered at the singular point, and confirm
that q and γ are the total flux rate and circulation, respectively.
2. For a two-dimensional point vortex at r = 0, show that ▽ × V = 0 and
▽ · V = 0 if r > 0.
3. Derive the Kutta-Joukowski theorem (15) for steady two-dimensional
flow, without linearizing the Bernoulli equation. Start with (4.90) and
assume that at large distances from the foil the velocity potential consists
of the combination of a free stream of velocity U and a point vortex of
circulation Γ.
Lifting Surfaces
243
4. Show that the complex velocity function u − iv= −(iΓ/2π) (z2 − 1)−1/2 satisfies the condition of zero normal velocity on a flat plate situated on the cut
(−1, 1), and behaves like a vortex of circulation Γ at large distances from the
origin. Why is this not the correct solution for a flat-plate hydrofoil?
5. Derive equation (41) for the nonlinear lift coefficient of a flat plate by
superposition of a flow tangential to the plate with velocity U cos a, a flow
of the form (4.58) normal to the plate, and a suitable amount of circulation
to satisfy the Kutta condition. The complex velocity for the latter component is given in problem 4.
6. What is the ideal angle of attack for a flat plate?
7. If the lift coefficient of a cambered two-dimensional hydrofoil is 0.8, at
α = 5 degrees, what is CL at α = 10 degrees?
8. According to the Weissinger “quarter-three-quarter-chord” approximation, the bound vortex distribution of a lifting surface can be approximated
by a single point vortex situated at the quarter-chord position x = l/4 with
circulation chosen to satisfy the boundary condition at the three-quarterchord position x = −l/4. Show from the Kutta-Joukowski theorem that the
resulting lift coefficient is exact for the two-dimensional parabolic meancamber line (43) and that the quarter-and three-quarter chord positions
form the only combination that gives the correct result in this case.
9. From the approximation described in problem 8 and the method of
images, show that a two-dimensional hydrofoil moving above a horizontal
plane will experience an increased lift force due to the “ground effect” of
the plane surface, whereas the interference between two foils with the same
angle of attack in a “biplane” configuration will decrease the lift force on
each.
10. Show that a uniform distribution of two-dimensional dipoles, oriented
in the vertical direction and distributed along the negative x-axis, is identical to a point vortex at the origin.
11. Find the lift and drag forces on a sailboat keel, assuming that the keel
profile is one half of an ellipse with high aspect-ratio and that the bottom
of the hull is a large flat plane coincident with the minor axis of the ellipse.
12. Write the complex potential of a pair of vortices, with opposite
strengths, separated by a distance 2a. A free vortex in a fluid is generally
assumed to be convected with the velocity of the fluid at the point where
244
Chapter 5
the vortex is situated. On this basis, what is the free motion of the vortex
pair discussed above? What is the free motion if the two vortices have the
same circulation?
13. A two-dimensional uncambered hydrofoil is pivoted about a point at
distance d downstream from the leading edge. For what values of d is the
hydrofoil statically stable at zero angle of attack? Discuss qualitatively the
situation that would result if the pivot is further downstream.
14. An uncambered three-dimensional lifting surface with aspect ratio
A = 5 experiences a lift coefficient of 0.4 at an angle of attack α = 5 degrees.
What is the minimum induced drag coefficient in this condition and at
α = 10 degrees?
15. A two-dimensional hydrofoil consists of a parabolic arc situated at an
angle of attack, as in equation (43). Compare the lift and drag forces of this
foil in the fully wetted and supercavitating conditions.
16. A hydrofoil set at an angle of attack of 7 degrees relative to the zero-lift
angle moves through a wave system such that the unsteady component of
the angle of attack is ±4 degrees. Compare the magnitudes of the mean and
unsteady lift forces, assuming the hydrofoil travels a distance of five chord
lengths in each period of oscillation.
References
Abbott, I. H., and A. E. von Doenhoff. 1959. Theory of wing sections. New York:
Dover.
Abramowitz, M., and I. A. Stegun, eds. 1964. Handbook of mathematical functions.
Washington: U.S. Government Printing Office.
Acosta, A. J. 1973. Hydrofoils and hydrofoil craft. Annual Review of Fluid Mechanics
5:161–184.
Barker, S. J., and T. M. Ward. 1976. Experiments on two- and three-dimensional cavitating hydrofoil models with and without flaps. California Institute of Technology, Guggenheim Aeronautical Laboratory report HSWT-1130.
Batchelor, G. K. 1967. An introduction to fluid dynamics. Cambridge: Cambridge University Press.
Bisplinghoff, R. L., H. Ashley, and R. L. Halfman. 1965. Aeroelasticity. Reading, Mass.:
Addison-Wesley.
Lifting Surfaces
245
Brard, R. 1972. Vortex theory for bodies moving in water. In Ninth symposium on
naval hydrodynamics, eds. R. Brard and A. Castera, pp. 1187–1284. Washington: U.S.
Government Printing Office.
Carrier, G. F., M. Krook, and C. E. Pearson. 1966. Functions of a complex variable. New
York: McGraw-Hill.
Cox, G. G., and W. B. Morgan. 1972. The use of theory in propeller design. Marine
Technology 9:419–429.
Hildebrand, F. B. 1976. Advanced calculus for applications. 2nd ed. Englewood Cliffs,
N. J.: Prentice-Hall.
Kerwin, J. E. 1973. Computer techniques for propeller blade section design. International Shipbuilding Progress 20:227–251.
Muskhelishvili, N. I. 1953. Singular integral equations. Groningen, Netherlands:
Nordhoff.
Robinson, A., and J. A. Laurmann. 1956. Wing theory. Cambridge: Cambridge University Press.
Thwaites, B., ed. 1960. Incompressible aerodynamics. Oxford: Oxford University Press.
Tulin, M. P. 1964. Supercavitating flows—small perturbation theory. J. Ship Res. 7
(3): 16–37.
Van Dyke, M. 1975. Perturbation methods in fluid mechanics. 2nd ed. Stanford:
Parabolic Press.
Woods, L. C. 1961. The theory of subsonic plane flow. Cambridge: Cambridge University Press.
Wu, T. Y. 1968. Inviscid cavity and wake flows. Basic Developments in Fluid Dynamics
2:1–116.
Wu, T. Y. 1971. Hydromechanics of swimming propulsion, Part 1. Journal of Fluid
Mechanics 46:337–355.
Wu, T. Y. 1972. Cavity and wake flows. Annual Review of Fluid Mechanics 4:243–284.
Wu, T. Y., A. K. Whitney, and C. Brennen. 1971. Cavity-flow wall effects and correction rules. Journal of Fluid Mechanics 49:223–256.
Wu, T. Y., C. J. Brokaw, and C. Brennen eds. 1975. Proceedings of the symposium on
swimming and flying in nature, vol. 2. New York: Plenum.
6 Waves and Wave Effects
Chapter
Waves
and
Wave
6
Effects
© Massachusetts Institute of TechnologyAll Rights Reserved
A unique aspect of marine hydrodynamics, relative to other branches of
fluid mechanics, is the importance of wave effects on the free surface. The
complexity of this topic is apparent from observations of the waves generated by a storm at sea, by a moving ship in calm water, or simply by
throwing a pebble into a pond. These are diverse examples of surface waves;
for ocean engineers and naval architects these are the most common and
important wave phenomena in the ocean.
Other wave phenomena that also exist in the ocean may be of comparable significance in special circumstances. Internal waves, like surface waves,
result from the balance of kinetic and potential energy, but they are found
in internal regions of density stratification beneath the sharp interface with
the atmosphere. Because of the small density differences involved, internal
waves occur at much lower frequencies, with periods on the order of several minutes. As a result, the influence of internal waves on bodies in the
ocean is generally negligible unless the body has an unusual low-frequency
resonance.
Waves of even lower frequencies exist, including inertial waves associated with the Coriolis acceleration due to the earth’s rotation, and tides
generated by changes in the potential energy of heavenly bodies. At the
high-frequency end of the spectrum are capillary waves and ripples, which
may be observed on the ocean surface and in small wave tanks, but these do
not affect large vessels or structures. A description of these different types of
wave motion from the oceanographic standpoint is given by Phillips (1966)
and by Neumann and Pierson (1966).
In marine hydrodynamics we are concerned with the effects of the wave
environment on floating or fixed structures. Of particular interest are the
wave loads on fixed structures and the oscillatory motions of vessels free to
248
Chapter 6
respond to the waves. Also of engineering importance are the waves generated by the motion of bodies in otherwise calm water, including the prediction of wave resistance for a moving ship.
These problems are amenable to a theoretical description based on the
assumptions that the fluid is ideal and that the wave motions are sufficiently small to linearize. Nonlinear effects are important in special circumstances, such as the breaking of waves in shallow water or locally near
a ship’s bow. Viscous stresses are significant for the wave forces on small
bodies, as noted in section 2.13, and in the generation of waves by wind.
Nevertheless, we shall find here, as in chapter 5, that useful results can be
obtained by neglecting nonlinear and viscous effects; in fact, these assumptions are practically essential if the complexity is to be kept in reasonable
bounds.
6.1 Linearized Free-Surface Condition
The fluid velocity V is expressed by the gradient of a velocity potential ϕ,
and the effects of the free surface must be expressed in terms of appropriate
boundary conditions on this surface. The physical nature of a free surface
requires both a kinematic and a dynamic boundary condition—that is, the
normal velocities of the fluid and of the boundary surface must be equal,
and the pressure on the free surface must be atmospheric. In both boundary
conditions the simplifications resulting from linearization are significant.
Similar assumptions were used in chapter 5 to derive the kinematic boundary condition on the wetted surface of a hydrofoil and the dynamic boundary conditions on both trailing vortex sheets and cavity surfaces.
We shall begin by stating the “exact” boundary conditions, assuming
only that the fluid is ideal and that surface tension at the free surface is
negligible (see section 2.3). The linearized results will follow by neglecting
second- and higher-order terms in the wave amplitude and associated fluid
motions.
A Cartesian coordinate system (x, y, z) is adopted, with y = 0 the plane
of the undisturbed free surface and the y-axis positive upward. The vertical
elevation of any point on the free surface may be defined by a function
y = η(x, z, t). In the special case of two-dimensional fluid motion,1 parallel
to the x-y plane, the dependence on z will be deleted.
Waves and Wave Effects
249
The exact kinematic boundary condition can be derived most readily by
requiring that the substantial derivative of the quantity y − η vanish on the
free surface. The result of this condition is that, on y = η,
0=
D
∂φ ∂η ∂φ ∂η ∂φ ∂η
( y − η) =
.
−
−
−
Dt
∂y ∂t ∂x ∂x ∂z ∂z
(1)
If the wave elevation η is sufficiently small, the slopes ∂η/∂x and ∂η/∂z
will be small quantities compared to one and of the same order of magnitude as ∂η/∂t. Similarly, if the fluid velocity is a small quantity proportional
to the wave motion, the derivatives of the velocity potential in (1) will be
small first-order quantities. Thus, the last two terms in (1) are of second
order and may be neglected in the linearized kinematic boundary condition, which is therefore given by
∂η ∂φ
.
=
∂t ∂y
(2)
This approximate boundary condition simply states that the vertical velocities of the free surface and fluid particles are equal, ignoring the small departures of that surface from the horizontal orientation. (A similar boundary
condition is given by equation (5.193) for the unsteady hydrofoil, but with
an additional contribution due to the free-stream velocity.)
The dynamic boundary condition is obtained from Bernoulli’s equation
(4.26). Assuming the atmospheric pressure pa is independent of position on
the free surface and choosing the constant of integration C(t) suitably, we
find that the exact condition to be satisfied on the free surface is
1
∂φ 1
− ( p − pa ) =
+ ∇φ ⋅ ∇φ + gy = 0.
ρ
∂t 2
(3)
Substituting the free-surface elevation η for y and solving for η,
η=−
1  ∂φ 1
+ ∇φ ⋅ ∇φ  .

g  ∂t 2
(4)
If we neglect the last term, which is a second-order quantity in the fluid
velocity, the linearized equation for the free-surface elevation is
η=−
1 ∂φ
.
g ∂t
(5)
250
Chapter 6
Strictly speaking, these boundary conditions should be imposed on
the exact free surface y = η. However, it is consistent with the linearizations already carried out to impose the first-order boundary conditions (2)
and (5) on the undisturbed plane of the free surface, y = 0. This additional
simplification of the linearization procedure can be justified formally by
expansion of the velocity potential and its derivatives in Taylor series about
the plane y = 0, as will be shown in section 6.4.
On the plane y = 0, the dynamic boundary condition (5) can be differentiated with respect to time and combined with the kinematic condition (2).
This gives a single boundary condition for the velocity potential,
∂ 2φ
∂φ
= 0, on y = 0.
+g
∂t 2
∂y
(6)
6.2 Plane Progressive Waves
The simplest solution of the free-surface condition (6), which nevertheless has great practical significance, is the plane progressive wave system.
This motion is two dimensional, sinusoidal in time with radian frequency
ω, and propagates with phase velocity Vp such that, to an observer moving with this velocity, the wave appears steady-state. Thus, the free-surface
elevation must be of the general form
η( x, t ) = A cos( kx − ωt + ε ),
(7)
where the positive x-axis has been chosen to coincide with the direction of
wave propagation. Here A is the wave amplitude, and ε is an arbitrary phase
angle which can be set equal to zero by suitable choice of the origin x = 0.
Hereafter this will be assumed, with ε = 0. The parameter
k = ω / Vp
(8)
is the wavenumber, the number of waves per unit distance along the x-axis.
Clearly,
k = 2π / λ ,
(9)
where the wavelength λ is the distance between successive points on the
wave with the same phase, as shown in figure 2.1.
The solution of this problem is expressed in terms of a two-dimensional
velocity potential ϕ(x, y, t), which must satisfy Laplace’s equation, the
Waves and Wave Effects
251
free-surface condition (6), an appropriate boundary condition on the bottom of the fluid, and must yield the wave elevation (7) from (5). Clearly
this potential must be sinusoidal in the same sense as (7); therefore we seek
a solution of the form
φ( x, y , t ) = Re[Y ( y )e − ikx + iωt ].
(10)
From Laplace’s equation, Y must satisfy the ordinary differential equation
d 2Y
− k 2Y = 0,
dy 2
(11)
throughout the domain of the fluid.
The most general solution of (11) is given in terms of exponential functions, in the form
Y = Ce ky + De − ky .
(12)
Here C and D are constants to be determined from the boundary conditions
on the free surface and bottom.
For now we assume that the fluid depth is infinite; hence (11) must hold
for −∞ < y < 0. To avoid an unbounded motion deep beneath the free surface, the constant D in (12) must vanish; thus
Y = Ce ky .
(13)
If this function is substituted in (10), the velocity potential is given by
φ = Re[ Ce ky −ikx + iωt ].
(14)
Use (5) to compute the linearized wave elevation, with y = 0, and compare
the result with the prescribed plane wave (7); then the constant C must
take the value
C = igA / ω .
(15)
The potential (14) can be rewritten in the form
φ=
gA ky
e sin( kx − ωt ).
ω
(16)
At this point in our derivation, the solution appears complete; yet the
free-surface condition (6) has not been imposed. Therefore (16) is too general, and if this potential is substituted in (6) an additional condition is
obtained, relating the wavenumber and the frequency in the form
252
Chapter 6
k = ω 2 / g.
(17)
Thus the frequency and wavenumber are mutually dependent parameters
related by the dispersion relation (17). Of course, the frequency ω can be
replaced by the wave period T = 2π/ω, just as the wavenumber k can be
replaced by the wavelength λ = 2π/k.
The phase velocity Vp can be determined from (8), and with (17) it follows that
Vp = ω / k = g / ω = ( g / k )1/2 = ( g λ / 2π )1/2 .
(18)
The last of these equivalent relations is consistent with the conclusion,
based on dimensional analysis in section 2.3, that the phase velocity is
proportional to the square root of the wavelength.
While the wave moves with the phase velocity Vp, the fluid itself moves
with a much smaller velocity given by the gradient of the potential (16).
The velocity components (u, v) of the fluid are
u=
∂φ
= ω Ae ky cos( kx − ωt ),
∂x
(19)
u=
∂φ
= ω Ae ky sin( kx − ωt ).
∂y
(20)
Since the horizontal and vertical velocity components have the same magnitude but a phase difference of a quarter period, the fluid particles move
through circular orbits of radius Aeky. On the linearized free surface y = 0,
the amplitude of this motion is equal to the wave amplitude A, in accordance with the kinematic boundary condition (2). Comparison of (19–20)
with the wave elevation (7) shows that the horizontal velocity component
is a maximum beneath the crest and trough. Beneath the crest the velocity
is positive, in the same direction as the wave propagation, and beneath the
trough the flow is in the opposite direction. The vertical velocity component is a maximum beneath the nodes η = 0, rising or falling with the free
surface. This velocity field is shown in figure 6.1.
Within the accuracy of the linear theory, the fluid particles move in
small circular orbits proportional to the wave amplitude; they remain in
the same mean position as the wave propagates through the fluid with a
phase velocity independent of the wave amplitude. Some nonlinear effects
that modify this situation will be discussed in sections 6.4 and 6.5, but
Waves and Wave Effects
253
Figure 6.1
Velocity field of a plane progressive wave in deep water. The phase velocity and
fluid velocity vectors are to the same scale. This example corresponds to the case
A/λ = 1/20.
for most practical purposes the linear results described here are extremely
accurate.
As the depth of submergence beneath the free surface increases, the fluid
velocity (19–20) is attenuated exponentially. For a submergence of half a
wavelength, ky = −π and the exponential factor is reduced to 0.04. Thus
waves in deep water are confined to a relatively shallow layer near the free
surface, with negligible motion beneath a depth of about 21 λ . On this basis
one can anticipate that if the fluid depth is finite, but greater than half a
wavelength, the effects of the bottom will be negligible. This conclusion
will be confirmed in section 6.3.
6.3 Finite-Depth Effects
For a fluid of constant depth h with the bottom a rigid impermeable plane,
the boundary condition
∂φ / ∂y = 0 ,
y = − h,
(21)
must be imposed. Returning to the general solution (12), both exponential
functions must be retained, with the constants C and D suitably chosen to
satisfy (21).
254
Chapter 6
The necessary algebra can be reduced by noting that the two independent solutions e±ky may be replaced by linear combinations of the hyperbolic sine and cosine function. The only combination that satisfies (21) is
proportional to cosh[k(y + h)]. Thus, for a fluid of finite depth,
Y(y ) = C
cosh k( y + h )
,
cosh kh
(22)
where the constant C has been chosen such that (15) remains valid. Substituting (22) in (10) and using (15) gives the velocity potential for finite
depth in the form
φ=
gA cosh k( y + h )
sin( kx − ωt ).
ω
cosh kh
(23)
Finally, for (23) to satisfy the free-surface condition (6), it follows that
k tanh kh = ω 2 / g .
(24)
The dispersion relation (24) is an implicit equation which determines
the wavenumber k and wavelength λ = 2π/k, for given values of the depth h
and frequency ω. Conversely, for given values of the wavelength and depth,
(24) can be used to calculate the frequency.
As the depth h tends to infinity, equations (23–24) reduce to the infinitedepth results (16–17) respectively. At a depth of
1
2
λ , the discrepancy
between (17) and (24) is less than half a percent, confirming the estimates
at the end of section 6.2.
The fluid velocity components can be computed as in (19–20), and for
finite depth it follows that
u=
∂φ gAk cosh k( y + h )
=
cos( kx − ωt ),
∂x
ω
cosh kh
(25)
v=
∂φ gAk sinh k( y + h )
=
sin( kx − ωt ).
∂y
ω
cosh kh
(26)
Once again there is a phase difference of a quarter period between these
two velocity components, but because the magnitudes are unequal, the trajectories of fluid particles will be elliptical. The major axis of these ellipses
is horizontal, and the velocity distribution is as shown in figure 6.2. Figure
6.7(a) is a time-exposure photograph that shows these trajectories vividly.
Waves and Wave Effects
255
Figure 6.2
Velocity field of a plane progressive wave in finite depth. The phase velocity and
fluid velocity vectors are to the same scale. This example corresponds to the case
A/λ = 1/20 and A/h = 1/5.
From (8) and (24) the phase velocity for finite depth can be expressed
in the form
Vp = ω / k = [( g / k )tanh kh ]1/2 .
(27)
This tends to the deep-water limit (18) for kh ≫ 1. The opposite limit,
kh ≪ 1, is the regime of shallow-water waves. In this case (27) can be approximated using the Taylor series for the hyperbolic tangent, and the leadingorder approximation for the phase velocity is
Vp ( gh )1/2 ,
kh 1.
(28)
The ratio of the phase velocity (27) to the infinite-depth limit (18) is plotted in figure 6.3, together with the depth-wavelength ratio h/λ. For a fixed
value of ω, it follows from (8) and (9) that Vp/Vp∞ = λ/λ∞. Both ratios increase
monotonically from zero to one, with increasing depth. If h > 21 λ , the phase
velocity and wavelength are essentially equal to their infinite-depth values.
Dimensional plots of the phase velocity, period, and wavelength are given
by Wiegel (1964) and Le Méhauté (1976).
In general, the phase velocity (27) depends on the wavenumber and the
wavelength. Thus water waves are dispersive, as noted in section 2.3, with
long waves traveling faster than shorter waves. The shallow-water limit (28)
is an exception, where the phase velocity depends only on the depth, and
the resulting wave motion is nondispersive.
256
Chapter 6
Figure 6.3
Phase velocity, wavelength, and depth ratios of a plane progressive wave. The deepwater limiting values Vp∞ and λ∞ can be determined from equation (18).
For a fluid of nonuniform depth, these results can be applied locally,
provided the change of depth is small over distances comparable to a wavelength. If waves of constant frequency propagate from deep to shallow
water, as at a beach, it follows from figure 6.3 that the wavelength and
phase velocity will decrease. Figure 6.4 illustrates this phenomenon in a
small wave tank.
6.4 Nonlinear Effects
In this section we shall consider briefly some of the simpler effects neglected
in the linearized theory. First the nonlinear free-surface boundary condition
Waves and Wave Effects
257
Figure 6.4
Sequence of photographs, taken through the transparent side of a wave tank, showing the propagation of plane progressive waves into shallow water. The water is darkened with dye, and the waves are generated by an oscillating vertical wedge at the left
side of the tank. Each wave crest is connected in successive photographs by diagonal
lines, which advance in time with the phase velocity. Both the phase velocity and
wavelength are reduced as the waves propagate toward the shallow end of the tank.
The interval between successive photographs is 0.25s, and the wave period is 0.4s.
The water depth is 0.11m at the left end, decreasing to zero at the right. Nonlinear
distortion occurs when the depth is comparable to the wave height.
must be derived to replace (6). This can be done, starting with the exact
kinematic and dynamic boundary conditions (1) and (4), but the necessary
derivatives of (4) are difficult to evaluate since this equation holds only on
the curvilinear surface η(x, z, t).
A more expedient approach is to replace the kinematic condition (1)
by the statement that the substantial derivative of the pressure is zero on
the free surface. This is a rather pragmatic mixture of the dynamic and
kinematic boundary conditions, since the statement that Dp/Dt = 0 on the
free surface implies that this is precisely the appropriate moving surface on
which the pressure is constant. Substituting (3) for the pressure, we obtain
the desired boundary condition in the form
∂
∂φ 1
0 =  + ∇φ ⋅ ∇  
+ ∇φ ⋅ ∇φ + gy  .
 ∂t
  ∂t 2

(29)
Working out the indicated derivative gives
∂ 2φ
∂φ
∂φ 1
+g
+ 2∇φ ⋅ ∇
+ ∇φ ⋅ ∇(∇φ ⋅ ∇φ ) = 0.
∂t 2
∂y
∂t 2
(30)
The free-surface boundary condition (30) is exact and explicit, except
that it must be applied on the unknown surface y = η defined by (4). If only
the linear terms are retained in (30), this equation reduces to (6).
258
Chapter 6
A systematic procedure can be used to expand the boundary condition
(30), as well as (4), from the exact free surface to the plane y = 0, using
Taylor series expansions of the potential and its derivatives. A typical
example is
1  ∂ 2φ 
 ∂φ 
φ( x, η, z , t ) = φ( x, 0, z , t ) + η   + η2  2  + ..
 ∂y  y = 0 2  ∂y  y = 0
(31)
Using this scheme for each of the derivatives in (4) and (30), we can obtain
a sequence of boundary conditions valid on the known surface y = 0. The
first three members are
∂ 2φ
∂φ
+g
= 0 + O(φ 2 ),
∂t 2
∂y
(32)
∂ 2φ
∂φ
∂ φ 1 ∂ φ ∂  ∂ 2φ
∂φ 
+g
+ 2∇φ ⋅ ∇
−
+ g  = 0 + O(φ 3 ),,
2
∂t
∂y
∂t g ∂t ∂y  ∂t 2
∂y 
(33)
∂ 2φ
∂φ
∂φ 1
+g
+ 2∇φ ⋅ ∇
+ ∇φ ⋅ ∇(∇φ ⋅ ∇φ )
∂t 2
∂y
∂t 2
−
∂φ 
1 ∂ φ ∂  ∂ 2φ
∂φ
+ 2∇φ ⋅ ∇ 
+g
g ∂t ∂y  ∂t 2
∂t 
∂y
−
∂φ 
1  1 ∂ φ ∂ 2φ 1
 ∂  ∂ 2φ
−
+g 
+ ∇φ ⋅ ∇φ 

 ∂y  ∂t 2
∂y 
g  g ∂t ∂y ∂t 2
+
1  ∂ φ  2 ∂ 2  ∂ 2φ
∂φ 
+ g  = 0 + O(φ 4 ).
2 g 2  ∂t  ∂y 2  ∂t 2
∂y 
(34)
The symbol O() indicates the order of magnitude of the neglected terms.
Equation (34) can be simplified by invoking Laplace’s equation and (32)
to eliminate the last group of terms. Nevertheless, the complexity of this
systematic procedure increases rapidly with the order of accuracy of the
boundary condition.
Plane progressive waves in deep water are of particular interest and
importance, and equations (32–34) can be utilized to obtain the third-order
expansion for plane waves.
Direct substitution of the first-order plane-wave potential (16) in the
second-order boundary condition (33) reveals that the second-order terms
in (33) vanish. Therefore the first-order potential is a solution of the secondorder boundary value problem, and we can state that
Waves and Wave Effects
φ=
gA ky
e sin( kx − ωt ) + O( A3 ).
ω
259
(35)
On the other hand, the free-surface elevation must be corrected for secondorder effects, using the systematic expansion of (4) in the form
η=−
=−
1  ∂φ 1
+ ∇φ ⋅ ∇φ 
 y =η
g  ∂t 2
1  ∂φ 1
+ ∇φ ⋅ ∇φ 
 =0
g  ∂t 2
+η
=−
∂  1  ∂φ 1

+ ∇φ ⋅ ∇φ   + −
 y = 0
∂y  g  ∂t 2
(36)
1  ∂φ 1
1 ∂ φ ∂ 2φ 
+ O(φ 3 ).
+ ∇φ ⋅ ∇φ −

g  ∂t 2
g ∂t ∂y ∂t  y = 0
Combining equations (35) and (36) yields the following second-order
results:
η = A cos( kx − ωt ) − 21 kA2 + kA2 cos2 ( kx − ωt )
= A cos( kx − ωt ) + 21 kA2 cos 2( kx − ωt ).
(37)
The last term in (37), which represents the second-order correction to the
free-surface profile, is positive both at the crests (kx − ωt) = 0, 2π, 4π, … and
at the troughs (kx − ωt) = π, 3π, 5π, … . The crests will be steeper, and the
troughs flatter, as a result of this nonlinear correction.
For the third-order free-surface condition (34), substituting the planewave potential (35) in the nonlinear terms in (34) eliminates all but one
term; thus the boundary condition for the third-order plane-wave solution
is given by
∂ 2φ
∂φ 1
+g
+ ∇φ ⋅ ∇(∇φ ⋅ ∇φ ) = 0 + O(φ 4 ).
∂t 2
∂y 2
(38)
The first-order solution (16) will satisfy this third-order boundary condition
provided the dispersion relation (17) is corrected for a second-order effect
of the form
ω 2 = gk(1 + k 2 A2 ) + O( k 3 A3 ).
(39)
The relation (8) can be utilized with (39) to obtain the corresponding correction for the phase velocity,
260
Chapter 6
Vp = ω / k = ( g / k )1/2 (1 + k 2 A2 )1/2 + O( k 3 A3 )
= ( g / k )1/2 (1 + 21 k 2 A2 ) + O( k 3 A3 ).
(40)
Thus the phase velocity depends weakly on the wave amplitude, and waves
of large height travel faster than small waves. By analogy with the more significant dependence of the phase velocity on wavelength, this dependence
on amplitude is known as amplitude dispersion.
If (39) is used to replace the linear dispersion relation (17), the linearized velocity potential (16) and (35) is accurate up to and including terms
of order A3—that is, the error in this simple velocity potential is O(A4).
The absence of second- and third-order terms in the potential holds
only for deep water, however. The nonlinear corrections for finite depth,
including the effects of surface tension, are given by Wehausen and Laitone
(1960).
Nonlinear solutions for plane waves based on systematic power series in
the wave amplitude are known as Stokes’ expansions. The most extensive of
these is due to Schwartz (1974), who utilized a digital computer to carry out
the infinite-depth expansion to order A117 and the finite-depth results to a
more modest point. Ultimately one might be concerned with the convergence of these power series. In 1918–1924 it was established by Levi-Civita
and Struik that these power series converged, at least for sufficiently small
wave amplitudes. Almost fifty years later it has been shown that nonlinear
plane waves are unstable if they are contaminated by side-band waves of
slightly different frequencies.
Thus, in practice, it is impossible to propagate a plane wave system in
a long wave tank, and waves that appear stable near the wavemaker will
deteriorate completely after traveling several hundred wavelengths. Further
details of this and related topics are contained in Lighthill (1967).
A complementary approach to nonlinear waves has been the study of
the highest wave. It can be shown that the limiting form for a plane wave
system has a sharp crest, with a local stagnation point at the crest, and with
the free surface inclined 30 degrees from the horizontal on each side of the
crest. In deep water this will occur when the ratio of wave height to wavelength is approximately 0.14, or about 1/7, as shown in figure 6.5. Details of
this analysis are summarized by Wehausen and Laitone (1960).
A description of nonlinear progressive waves was first derived in 1802 by
Gerstner, who showed that the free-surface boundary conditions could be
Waves and Wave Effects
261
Figure 6.5
The “steepest wave” profile.
satisfied exactly by a trochoidal2 wave profile. This trochoidal wave system
has been used extensively in the field of naval architecture, but it is scientifically deficient since the velocity field is not irrotational. The practical
appeal of the trochoidal wave results from the similarity of its appearance
to actual waves, but the same is true of the second-order Stokes wave profile
(37), which actually agrees with the second-order approximation of a trochoid! Descriptions of the trochoidal wave system may be found in Lamb
(1932), as well as in Wehausen and Laitone (1960). Wehausen and Laitone
note that while the occurrence of vorticity in ocean waves cannot be ruled
out, due to viscous effects, the sign of vorticity in Gerstner’s wave is opposite that expected from the action of shear stresses due to wind blowing
over the free surface. In any event, the mechanism of wave generation by
wind involves more complex processes than a simple shear stress on the
interface, as discussed by Phillips (1966).
6.5 Mass Transport
One of the more interesting nonlinear features of plane progressive waves
is the occurrence of a second-order mean drift of the fluid particles, in the
same direction as the wave propagation. This effect can be calculated most
easily for the infinite-depth case, since the velocity potential (16) is exact
to second order in the wave amplitude. The existence of a net flux follows
because the horizontal velocity component (19) is equal in magnitude and
opposite in sign beneath the crest and trough, at points of equal depth y
below the mean free surface. Since u is positive beneath the crest, where the
total elevation of the fluid is greater, the total horizontal flux beneath the
262
Chapter 6
crest will exceed that beneath the trough, and on the average a net mass
transport will occur.
The orbital motion of a particular fluid particle can be computed in
terms of the Lagrangian coordinates [x0(t), y0(t)] which define the position
of a particle. These must satisfy the relation s
dx0
= u( x0 , y 0 , t ),
dt
(41)
dy 0
= v( x0 , y 0 , t ).
dt
(42)
If x0 and y0 differ by a small amount, of order A, from the fixed position
(x, y), Taylor series can be used to expand (41–42) in the form
dx0
∂u
∂u
= u( x, y , t ) + ( x0 − x)
+ ( y0 − y )
+ O( A3 ),
dt
∂x
∂x
(43)
dy 0
∂v
∂v
= v( x, y , t ) + ( x0 − x) + ( y 0 − y ) + O( A3 ).
dt
∂x
∂y
(44)
Integrating (19) and (20) with respect to time gives the first-order trajectories
( x0 − x) = ∫ u dt = − Ae ky sin( kx − ωt ) + O( A2 ),
(45)
( y 0 − y ) = ∫ v dt = Ae ky cos( kx − ωt ) + O( A2 ).
(46)
If these expressions are substituted with (19–20) in (43–44), it follows
that
dx0
= ω Ae ky cos( kx − ωt ) + ω kA2 e 2 ky + O( A3 ),
dt
(47)
dy 0
= ω Ae ky sin( kx − ωt ) + O( A3 ).
dt
(48)
The second-order vertical motion of a given particle of fluid is strictly
periodic, with the velocity given by (48), but the horizontal velocity (47)
contains a steady Stokes’ drift. Integrating (47) vertically with respect to all
particles of fluid −∞ < y0 < 0 yields the total mean flux
Q = 21 ω A2 = 21 kA2Vp .
(49)
Waves and Wave Effects
263
The same result could be derived directly from a computation of the average flux across a vertical column of fluid −∞ < y < η (see problem 3).
The presence of a mean drift is obvious from the observation of small
vessels floating in waves, although here the mean motion of the vessel may
be affected more by wind forces and other nonlinear effects such as surfing. Mass transport can result in the piling up of water at a beach, with an
associated increase in the local mean depth. In some cases a long-shore current is associated with this phenomenon. This effect is discussed by Wiegel
(1964). In shallow water, viscous effects significantly modify the mean drift
flow, in a manner described by Longuet-Higgins (1953) and by Ünlüata and
Mei (1970).
6.6 Superposition of Plane Waves
The plane progressive wave described in sections 6.2 and 6.3 is a single,
discrete wave system, with a prescribed monochromatic component of
frequency ω and wavenumber k, moving in the positive x-direction. This
wave system can be generalized in several respects. First, the wave profile
(7) rewritten in the form
η = A cos( kx + ωt ),
(50)
corresponds to a plane progressive wave moving in the negative x-direction.
The subsequent relations derived in sections 6.2 and 6.3 are unchanged,
except that the phase function (kx − ωt) is replaced by (kx + ωt) throughout,
and the signs of some expressions must be reversed.
More generally, in three dimensions, a plane wave moving in an arbitrary direction, at a polar angle θ in the horizontal x-z plane, can be analyzed by rotating the coordinate system appropriately. As shown in figure
6.6, the coordinate x′ may be defined to coincide with the direction of wave
propagation. Since x′ = x cosθ + z sinθ, the appropriate generalization of (7)
is given by
η = A cos( kx cos θ + kz sin θ − ωt ),
(51)
with similar expressions for the velocity potentials (16) and (23). Note that
for θ = π, (51) reduces to (50).
In the linearized theory, solutions may be superposed without violating
the boundary conditions or Laplace’s equation. Thus considerable scope
264
Chapter 6
Figure 6.6
View from above of a plane progressive wave system, moving parallel to the x′-axis at
an angle θ relative to x. The wave crests are shown by dashed lines.
for further generalizations is provided by the superposition of plane-wave
solutions. The simplest example is a standing wave formed by adding two
identical plane waves moving in opposite directions. The sum of these two
wave systems is
η = A cos( kx − ωt ) + A cos( kx + ωt )
= 2 A cos kx cos ωt ,
(52)
and the velocity potential for deep water is
φ=−
2gA ky
e cos kx sin ωt .
ω
(53)
In finite depth the corresponding result is obtained without difficulty
from (23).
The standing wave (52) is sinusoidal in time, for fixed position x, and
vice versa. This wave motion is therefore oscillatory but not progressive,
as shown in figure 6.7(c); it is typical of the “sloshing” motions in closed
containers such as swimming pools, tanks, and the wells of some drilling or
research ships. If the container is rectangular of width w, with vertical walls
at x = 0 and x = w, the boundary conditions of zero horizontal velocity on
Waves and Wave Effects
265
Figure 6.7
Particle trajectories in a plane progressive wave (a), a partial reflected wave (b), and a
standing wave (c). These correspond respectively to a reflection coefficient of 0, 0.38,
and 1.0 in equation (56). Note that the reflection coefficient can be measured from
the maximum and minimum of the envelope, using (56). These photographs are
based on time exposures, and are reproduced from a more extensive series of observations made by Ruellan and Wallet (1950).
these walls will be satisfied by (53) if kw = nπ, with n any integer. Therefore
a discrete spectrum of eigenfrequencies exist where standing waves are resonant in such a container.
Standing waves are also of physical relevance if the plane waves are incident upon a perfectly reflecting vertical wall, say at x = 0, and the solution
(52–53) corresponds to the two-dimensional case where the wave crests are
parallel to the wall. If the fluid domain is x < 0, the first term in (52) is the
incident wave and the second term is the reflected wave. For oblique waves,
the appropriate generalization follows from (51) in the form
266
Chapter 6
η = A cos( kx cos θ + kz sin θ − ωt ) + A cos( kx cos θ − kz sin θ + ωt )
= 2 A cos( kx cos θ )cos( kz sin θ − ωt ).
(54)
Note that the corresponding potential will satisfy the boundary condition
of zero normal velocity on the wall x = 0. The reflected wave that has
been added in (54) is consistent with Snell’s law and with the method of
images.
If plane waves are incident upon a beach, which can absorb a portion
of the incident wave energy, or upon a vessel, which reflects part of the
wave energy and transmits the remainder, the amplitude of the reflected
wave will be reduced. In this case the reflected and incident waves in (52)
or (54) will differ, and their ratio defines a reflection coefficient R. In the
two-dimensional case, with the incident wave (7) traveling from −∞, (52)
is replaced by
η = A Re[ e − ikx + iωt + Reikx + iωt ].
(55)
Here a complex notation has been used since the phase of the reflected
wave may be nonzero, with the corresponding coefficient R complex. For a
beach, the magnitude of the reflection coefficient is generally small.
The presence of small reflected waves in a wave tank is generally apparent in terms of the amplitude modulation of the wave system. Thus, (55)
can be rewritten in the form
η = A Re[ e − ikx + iωt (1 + Re2 ikx )].
(56)
The factor in parenthesis is the amplitude variation, which varies in x with
twice the wavenumber or half the wavelength. This structure usually can
be noted, by observing the change in amplitude of a particular wave crest.
Figure 6.7(b) shows an exaggerated example. The reflection coefficient can
be measured by recording the wave amplitude at two or more positions and
analyzing the measured data in conjunction with (56).
More general wave motions, which are no longer monochromatic, can
be obtained by superposing plane waves of different wavenumbers or frequencies. In the two-dimensional case, we begin by forming a discrete sum
of waves of the form
N
η = ∑ Re[ An exp( −ikn x + iω nt )],
n =1
(57)
Waves and Wave Effects
267
with a corresponding representation of the velocity potential (16) or (23)
Here kn denotes the wave number of the nth wave component, and ωn its
frequency. These two parameters must satisfy the appropriate dispersion
relation (17) or (24) for each value of n. The amplitude An of each component is arbitrary and may be chosen to suit the desired solution, including
the possibility that each An is complex with a different phase.
If the total number of discrete waves N tends to infinity, while the difference between adjacent frequencies and wave numbers reduces to zero,
the sum (57) will tend to an integral over the continuous spectrum of
frequencies,
∞
η( x, t ) = Re ∫ A(ω )exp[ −ik(ω )x + iωt ] dω .
(58)
−∞
Here k(ω) > 0 must satisfy the appropriate dispersion relation (17) or (24);
negative values of the frequency have been admitted in this representation, to allow for the possibility of plane waves moving in the negative
x-direction, without explicitly writing out a second term of the form (50).
Equation (58) can alternatively be expressed in an equivalent form as an
integral over all possible wave numbers −∞ < k < ∞, with ω(k) > 0 determined
by the appropriate dispersion formula.
These distributions of wave systems can be extended from two to three
dimensions by introducing the oblique wave (51) and summing or integrating over all possible wave directions θ. In the most general case of a
continuous distribution, a two-dimensional integral representation follows
of the form
∞
2π
0
0
η( x, z , t ) = Re ∫ dω ∫ dθ A(ω ,θ )exp[ −ik(ω )( x cos θ + z sin θ ) + iωt ].
(59)
This expression will be used to analyze ship waves in section 6.10. It also
can be utilized, with A(ω, θ) a random variable, to represent a spectrum of
ocean waves. The latter application will be discussed in section 6.20.
The integral in (58) is recognized as a Fourier integral, and (59) is a twodimensional integral. Thus a large class of physically relevant wave systems
can be represented, and the inversion formulas from Fourier theory can be
used to find the appropriate amplitude functions A(ω) or A(ω, θ) required
to generate a prescribed wave elevation η. This procedure can be utilized
to solve the Cauchy-Poisson problem for the wave motion resulting from a
268
Chapter 6
transient initial disturbance such as would result by throwing a pebble into
a pond. The solution of this interesting problem is derived by Lamb (1932),
Stoker (1957), and Wehausen and Laitone (1960).
6.7 Group Velocity
Distributions of wave systems, such as those represented by the sum (57)
or the integrals (58–59), are difficult to describe in simple terms. Since the
component waves each travel with a different phase velocity, a continuously changing wave pattern results. Nevertheless, if we consider a narrow
band of the component waves, with nearly equal wavelength and direction, a characteristic of the resulting distribution is that the waves travel
in a group. The group velocity Vg can be derived from a dynamic analysis of
energy flux, but a simpler approach to this subject follows from a purely
kinematic study of the group of waves formed by two nearly equal plane
waves.
Two adjacent components of the discrete spectrum (57) can be written
in the form
η = Re{ A1e − ik1x + iω1t + A2 e − ik2 x + iω2t }
{
}
A


= Re A1e − ik1x + iω1t 1 + 2 e − iδ kx + iδωt  ,
A1


(60)
where δk = k2 − k1 and δω = ω2 − ω1. Here, as with the partial reflected
wave in (56), the factor in brackets represents an amplitude modulation;
but unlike (56), this factor will be slowly varying in space and time. The
wave form (60) is similar to the beat-frequency effect in electromagnetic
wave motions, where one refers to the first exponential factor in (60) as the
carrier wave.
This type of wave motion is illustrated in figure 6.8, which shows a group
of carrier waves enclosed by a slowly-varying envelope. The wavelength
and period of the carrier are the usual parameters λ = 2π/k1 and T = 2π/ω1.
By analogy, the wavelength of the group is 2π/δk, and its period in time is
2π/δω. Of particular interest is the group velocity Vg, given by the ratio
Vg =
δω
.
δk
(61)
Waves and Wave Effects
269
Figure 6.8
Wave group resulting from the superposition of two nearly equal plane waves as in
(60). The individual waves travel with the phase velocity, while the envelope travels
with the group velocity (61).
In the limiting case where δω → 0, δk → 0, (60) will tend to a monochromatic plane wave, but if x and t are large, the amplitude modulation
in (60) will persist so long as the products xδk and tδω are finite. Under this
circumstance the group velocity (61) will approach the finite limit
Vg =
dω
,
dk
(62)
in accordance with the classical definition of a derivative. Thus, a group of
waves of nearly equal frequency and wavenumber will propagate with the
velocity (62), as opposed to the phase velocity Vp = ω/k of the individual
wave components in the group.
In general, the phase and group velocities differ, unless ω/k = dω/dk,
which will occur only if the frequency and wavenumber are directly proportional and the phase velocity is constant. This exception occurs in the
shallow-water limit kh ≪ 1.
In deep water, the group velocity can be evaluated by solving the dispersion relation (17) for ω(k) and substituting this in (62). Thus
Vg = 21 ( g / k )1/2 = 21 ω / k = 21 Vp ,
(63)
and the group velocity is precisely half of the phase velocity. In finite depth,
the dispersion relation (24) can be combined with (62) to give
kh 
1
Vg =  +
V.
 2 sinh 2kh  p
(64)
This expression reduces to (63) if kh is large, whereas in the shallow-water
limit the factor in parenthesis in (64) reduces to unity. The value of the
270
Chapter 6
group velocity is plotted in figure 6.9, as a ratio of the infinite-depth phase
velocity. This figure shows that the group velocity is less than the phase
velocity; their ratio decreases from one to one-half as the water depth
increases. The curve in figure 6.9 also shows that for fixed frequency and
increasing depth, the group velocity attains a maximum value somewhat
greater than its infinite-depth limit, at an intermediate depth where h/λ is
about 0.2.
That individual waves move faster than the group implies that in a
wave system with a front, propagating into otherwise calm water, the
individual waves will overtake the front and vanish. This phenomenon
can be observed in a wave tank by starting the wavemaker from rest at a
Figure 6.9
Phase velocity and group velocity ratios for a plane progressive wave as a function
of the depth.
Waves and Wave Effects
271
given frequency so that a group of waves will propagate down the tank as
shown in figure 6.10. The front of the group moves at a slower velocity
than the individual wave crests behind it, and these crests appear to vanish
as they approach the front. After the wavemaker is turned off, the “back”
of the wave system moves with the group velocity; thus “new” waves that
move with the phase velocity must arise from calm water at the back. At
first glance it appears that the energy within the wave system is destroyed
in some sense, contrary to the principal of conservation of energy. In
fact, there is no contradiction, since the wave energy moves not with
the phase velocity but with the group velocity. To confirm this statement
requires an analysis of the wave energy and an alternative approach to the
group velocity based upon the rate of energy flux in a plane progressive
wave system.
6.8 Wave Energy
Water waves are the result of a balance between kinetic and potential energy
in the fluid, and the total energy is the sum of these two components. Thus,
in a prescribed volume , the total energy is given by the integral
1
KE + PE = ρ ∫∫∫  V 2 + gy  d  .
2

(65)

First, we will focus attention on a vertical column, extending throughout
the depth of the fluid and bounded above by the free surface; the energy
density E, per unit area of the mean free surface above this column, is
given by
η
η
1
1
1
E = ρ ∫  V 2 + gy  dy = ρ ∫ V 2 dy + ρ g (η2 − h2 ).
2

2
2
−h
−h
(66)
The potential energy − 21 ρ gh2 of the fluid below the equilibrium plane
y = 0 is of no interest, since this is a constant unrelated to the wave motion.
(Moreover, this term is a source of embarrassment in the limit h → ∞!) Therefore this contribution to (66) is omitted from the subsequent equations.
For plane progressive waves in infinite depth, the energy density can be
computed using the results of section 6.2. Substituting (19–20) in the last
integral in (66) and carrying out this integration over the depth yields
272
Chapter 6
Figure 6.10
Sequence of photographs showing a plane progressive wave system advancing into
calm water. The water is darkened with dye, and the lower half of the water depth
is not shown. The wave energy is contained within the heavy diagonal lines, and
propagates with the group velocity. (The boundaries of the wave group diffuse slowly
with time, due to dispersion.) The position of one wave crest is connected in successive photographs by the light line, which advances with the phase velocity. Each
wave crest moves with the phase velocity, equal to twice the group velocity of the
boundaries. Thus each wave crest vanishes at the front end and, after the wavemaker
is turned off, arises from calm water at the back. The interval between successive
photographs is 0.25s and the wave period is 0.36s. The wavelength is 0.23m and the
water depth is 0.11m.
Waves and Wave Effects
E=
ρω 2 A2 2 kη 1
e + ρ g η2 .
4k
2
273
(67)
For small-amplitude waves, the energy density (67) is proportional to the
square of the wave amplitude A, and to this order the exponential factor
in (67) may be set equal to one. Equivalently, (66) shows that the kinetic
energy is O(A2), and the contribution to the last integral between y = 0 and
y = η will give a higher-order contribution proportional to A3. Using the
dispersion relation (17) in the first term of (67) and substituting (7) for the
free-surface elevation we obtain
E = 14 ρ gA2 + 21 ρ gA2 cos2 ( kx − ωt ).
(68)
The first and second terms on the right side of (68) represent the kinetic
and potential energies, respectively. If these are averaged with respect to
time over one cycle of the wave motion, their contributions are identical,
and the total mean energy density is given by the expression
E = 21 ρ gA2 ,
(69)
where a bar denotes the time average.
For finite depth the corresponding relations can be derived with
somewhat more effort. The expression for the potential energy in (68) is
unchanged, and the mean value of the kinetic energy is identical to the first
term in (68). Thus (69) holds in general, along with the conclusion that the
mean energy is divided equally between kinetic and potential contributions
(see problem 4).
To analyze the rate of energy flux, we consider the rate of change with
respect to time of the total energy (65). Allowing the boundary surface S of
the volume  to move with the velocity Un and using the transport theorem
(3.11) gives
d
dE
1
∂ 1
= ρ ∫∫∫  V 2 + gy  d  = ρ ∫∫∫  V 2 + gy  d 


dt
dt   2
t 2
∂

1
+ ρ ∫∫  V 2 + gy  U n dS.
2

(70)
S
Since the vertical coordinate y is independent of time, the only contribution to the integrand of the last volume integral is from the kinetic energy
term, which takes the form
274
Chapter 6
∂ 1
∂φ
∂φ
∂φ 2
∇φ ⋅ ∇φ  = ∇φ ⋅ ∇
= ∇ ⋅  ∇φ  −
∇ φ.

 ∂t
 ∂t
∂t
∂t  2
(71)
The last term vanishes by Laplace’s equation, and the remaining contribution to the volume integral can be evaluated by the divergence theorem.
Thus
∂ 1
∫∫∫ ∂t  2 V

2
 d  = ∂φ ∂φ dS.
∫∫ ∂t ∂n

S
(72)
Substituting this result in (70) gives the time rate of change of energy in
the form
{
}
dE
∂φ ∂φ  1 2
= ρ ∫∫
+ V + gy  U n dS.

dt
∂t ∂n  2
S
(73)
Substituting Bernoulli’s equation (3), we can recast (73) in the alternative
form
dE
 ∂φ ∂φ  p − pa ∂φ  
= ρ ∫∫ 
−
+  U n  dS.
dt
∂t ∂n  ρ
∂t  
S 
(74)
Once again we restrict our attention to a volume of fluid, with vertical
sides, bounded above by the free surface and below by the bottom. Physically no contribution to (74) from the last two surfaces is anticipated, and
this can be confirmed by noting that ∂ϕ/∂n = Un = 0 on the bottom, whereas
on the free surface ∂ϕ/∂n = Un and p = pa. Thus the energy flux occurs across
the vertical control surface bounding the fluid column.
Equation (74) holds with Un an arbitrary normal velocity of the vertical
surface of the column. If the column moves with constant velocity U, in
the direction of the wave propagation, energy must enter one side of the
column at the same mean rate as it leaves on the other, as shown in figure
6.11. Thus the mean rate of energy flux across any vertical control surface
x = constant, per unit width in the z-direction, is given by
η
{
}
dE
∂φ ∂φ  1 2
= ρ∫
+ V + gy  U dy ,

dt
∂t ∂x  2
−h
(75)
where U is the velocity of this control surface in the + x-direction.
If (75) is set equal to zero, U must be equal to the mean velocity of the
energy flux in the fluid. On this basis it follows that
Waves and Wave Effects
275
Figure 6.11
Vertical control surfaces used to measure the rate of energy flux. Since the fluid domain bounded by these control surfaces is of constant width, the average energy
within this domain must be constant and the rate at which energy crosses each vertical surface must be equal to (75).
η
∂φ ∂φ
dy
∂t ∂x
−h
U = −ρ ∫
η
1
ρ ∫  V 2 + gy  dy .
2

(76)
−h
The denominator is the energy density (66), which can be evaluated from
(69). As in (66–68), the contribution of the interval 0 < y < η to the integral
in the numerator of (76) is of third order in A, and the leading-order contribution of order A2 can be computed with the upper limit set equal to zero.
On this basis, using the infinite-depth potential (16) gives
0
∂φ ∂φ
dy
∂t ∂x
−∞
U = −ρ ∫
0
2 gk
1
ρ gA2 =
∫ e 2 ky cos2 (kx − ωt )dy
2
ω −∞
(77)
= 21 g / ω = 21 Vp ,
where (18) has been used to substitute the phase velocity Vp. Comparing
this result with (63), we see that the energy’s velocity U is equal to the group
velocity Vg. The same conclusion holds in the general case of finite depth h,
as can be confirmed by substituting (23) for the potential in the numerator
of (77).
The mean rate of energy flux across a fixed control surface is the product
of the energy density and the group velocity,
dE
= Vg E.
dt
This result can be calculated directly from (75) with U = 0.
(78)
276
Chapter 6
Equation (78) can be used in conjunction with the energy-density (69)
to predict the change in height of a sinusoidal wave as it propagates through
a region of gradually changing depth. Provided the change in depth is small
over distances on the order of a wavelength, these relations can be applied
locally at each depth, and it can be assumed moreover that the reflected
energy is negligible. It then follows that the energy-flux (78) is constant
for all depths, and since the frequency ω remains constant, the local wave
height must be inversely proportional to the square root of the group
velocity. Referring to figure 6.9, the shoaling coefficient can be deduced as
the inverse square root of the group velocity shown by the dashed line.
Simultaneously, the wavelength and phase velocity will decrease with
depth (see the solid line in figure 6.9). Taking both effects into account, the
steepness of the waves will increase as the waves near the beach, leading
ultimately to wave breaking when the wave height is comparable to the
local depth.
This description of two-dimensional waves incident upon a beach is
based on the assumption of no energy reflection, which is valid provided
the bottom slope is small. Figure 6.4 illustrates this situation on a small
scale, with the bottom slope of the wave tank inclined at an angle of 2.5
degrees. The increasing steepness of the waves is apparent, but the change
in wave amplitude is not obvious. This can be explained from figure 6.9 by
noting that the shoaling coefficient exceeds l.0 only if the local depth is less
than 0.06 times the deep-water wavelength. Thus, for virtually the entire
length of the wave tank in figure 6.4, the shoaling coefficient is less than
one and the wave amplitude is attenuated.
This dynamic analysis of energy flux provides an alternative interpretation of the group velocity, complementary to that based on the kinematic
argument in section 6.7. From the physical viewpoint the equivalence of
these results is not surprising, particularly since each group of waves shown
in figure 6.8 is bounded by a region of relatively small wave amplitudes
and correspondingly small amounts of energy. If no energy is transferred
between adjacent groups, then the energy must travel with the velocity of
the group. A similar argument holds for the propagation of the front in figure 6.10, between a region of a calm water and a system of plane waves; in
this situation, no wave energy can be transferred across the front.
Since the group velocity is less than the phase velocity, the energy in a
plane progressive wave system will be propagated at a slower speed than
Waves and Wave Effects
277
the individual wave crests and will be “left behind” in the fluid. This seems
physically reasonable and explains why ship waves are observed downstream of the disturbance that generates them. There are exceptions to this
situation, however. For very short ripples governed by surface tension, the
group velocity exceeds the phase velocity and the resulting waves occur
upstream of the disturbance. This situation can be observed by slowly moving a suitable obstacle in a basin of water or by observing the waves generated by a calm stream flowing past a fish line. For the latter reason, this is
known as the fish-line problem; details may be found in Lamb (1932). Small
capillary waves will occur upstream of large bodies as well, but usually their
presence is obscured and of no practical significance.
Alternative derivations and interpretations of the group velocity exist.
A particularly elegant approach is given by Wehausen and Laitone (1960),
based on the propagation of a packet of wave energy in a narrow-band spectrum. An interpretation based on the kinematic analysis of a general slowlyvarying wave motion is emphasized by Whitham (1974). In section 6.11,
we shall find that the group velocity emerges in yet another, but related
sense, in conjunction with Kelvin’s method of stationary phase for approximating the far-field asymptotic form of wave motions.
6.9 Two-Dimensional Ship Waves
Having developed the analysis of wave energy radiation for plane progressive waves, we can now consider the problem of wave generation by moving vessels and the associated wave resistance. We begin with the simplest
case, the two-dimensional motion that would be generated by moving a
long cylindrical obstacle normal to its axis.
Since our approach will be based on energy conservation, we can ignore
the details of the flow near the body and focus attention on the wave system far downstream. If the body moves with constant velocity U in otherwise calm water and if the motion has reached a steady state, the only
waves that can exist downstream move with a phase velocity Vp = U. Any
other waves would either overtake the body, or drop further behind, in an
unsteady manner. Since the phase velocity is fixed, the wavelength and the
wave number must take prescribed values; for the deep-water case they are
given by λ = 2πU2/g and k = g/U2.
278
Chapter 6
The waves generated by the body contain energy that must be imparted
to the fluid as work done by the body on the fluid. Thus, the body will
experience a drag force D due to its wave resistance; the purpose of our
analysis is to relate D to the wave amplitude A far downstream.
Figure 6.12 shows a sketch of this situation in a fixed frame of reference,
as well as in a reference frame moving with the body. The fixed system
shown in (a) pertains to our analysis of the energy density and flux. The
values of these quantities are indicated in the figure in terms of the wave
amplitude A. The body and waves move with velocity U in the + x-direction,
and thus there is a positive flux of energy across the fixed control surface
downstream.
Since the control surface is fixed and the body is moving, the length of
the fluid region between the two will increase with velocity U, and the total
energy in this region will increase at a rate equal to the product of U and the
Figure 6.12
Two-dimensional ship waves as viewed by a fixed observer (a), and in a reference
frame moving with the ship (b).
Waves and Wave Effects
energy density
1
2
279
ρ gA2 . The energy input necessary to balance this increase
results in part from the work done at a rate DU by the body, in opposition
to the wave drag D. In accordance with (78), energy also enters the fluid
region across the control surface downstream, at a rate equal to the product
of the energy density and group velocity 21 U . From energy conservation it
follows that
1
2
ρ gA2U = DU + 14 ρ gA2U .
(79)
Solving this equation for the drag D yields
D = 14 ρ gA2 .
(80)
The same conclusion follows from the alternative analysis based on the
moving reference frame of figure 6.12(b). Here the position of the control
surface is fixed relative to the body, and the total energy of the intermediate fluid region is constant. The rate of work done by the body on the fluid
is unchanged by the transformation to a moving coordinate system, but
across the downstream control surface the energy now moves at the relative
velocity Vg − U = − 21 U . If the latter is multiplied by the energy density, the
energy balance takes the form
DU − 14 ρ gA2U = 0,
(81)
which is equivalent to (79) and (80).
Thus, if a two-dimensional body generates waves of amplitude A because
of its steady motion on the free surface, the associated wave drag is 14 ρ gA2.
This result by itself does not provide a means for predicting the wave
resistance, however, since we do not have a method for calculating the
wave amplitude A generated by the body. Physically, the latter quantity
will depend on the body geometry, as well as the velocity U. One might
expect that increasing the body size or the velocity will increase A and thus
the wave resistance. However, the latter conclusion overlooks the possibilities of destructive interference, which may be emphasized with a simple
example.
Let us consider a single small disturbance, which generates waves of
amplitude a. The resulting free-surface profile, in a coordinate system moving with the disturbance, will have the form
η = a cos( kx + ε ),
(82)
280
Chapter 6
where k = g/U2 and the phase angle ε is unimportant. From (80), the wave
resistance of this disturbance by itself is given by
1
4
ρ ga2 .
Next we consider the result of superposing a second disturbance, of
equal but opposite magnitude, situated at a distance l downstream of the
first. The total free-surface elevation resulting from this superposition of
two disturbances is
η = a cos( kx + ε ) − a cos( kx + ε + kl )
= Re{ae i( kx + ε ) (1 − e ikl )}.
(83)
The total wave amplitude downstream of both disturbances is
A = a 1 − e ikl = 2a sin( 21 kl ) ,
(84)
and the associated wave resistance is given by
D = ρ ga2 sin2 ( 21 kl ).
(85)
The amplitude (84) and drag (85) are oscillatory functions of the velocity
and length; the drag varies between zero and four times the corresponding
value for a single disturbance.
If the Froude number F = U/(gl)1/2 is introduced and (85) is plotted as
shown in figure 6.13, the importance of interference effects is obvious,
especially for the lower speeds. For spacing l equal to an integer number of
wavelengths, the waves and resulting wave resistance are zero, whereas for
spacings of an integer plus a half times the wavelength, the disturbance is
a maximum.
This simplified example can be related in practice to the bow and stern
wave systems of a more realistic vessel, particularly if the bow flow is represented by a point source and the stern by a point sink, as in the Rankine
bodies of chapter 4. To the extent that this analogy is valid it can be presumed that the length l is the distance between the source and sink, which
will be less than the overall length of the body. Similar interference effects
between the bow and stern exist in the three-dimensional case, but with
reduced magnitude, as we shall see in sections 6.12 and 6.13.
Analogous results follow if two positive disturbances are superposed.
Thus, if a pair of two-dimensional vessels moves in tandem, with one in
the other’s wake, the total wave resistance of the pair will fluctuate as in
figure 6.13 but with a shift of
1
2
λ in the definition of l. However, the drag
in question is the sum of the individual components acting on each vessel,
Waves and Wave Effects
281
Figure 6.13
Wave drag of a pair of two-dimensional disturbances. The strength of these is equal
and opposite, and their separation distance is l.
and vanishing of the total drag does not imply that each vessel experiences
no drag. What in fact occurs is that the upstream vessel experiences a drag,
and the downstream vessel an equal and opposite thrust force, much like a
surfboard riding on a wave system generated in this case by the upstream
vessel. Similar considerations apply to convoys of ships in three dimensions, but with reduced interference as noted above.
These simplified results are independent of the body geometry that produces the assumed waves and apply equally whether the body is floating
on the free surface or submerged. The simplest specific example, where the
wave amplitude A and wave resistance are related explicitly to the body
shape and speed, is a submerged circular cylinder; an approximate theory
for this case is outlined by Lamb (1932). The case of a planing body is of
particular interest; in many respects it resembles the flow beneath a lifting
surface, but with the added complications of the free surface. More extensive results for a wide variety of floating and submerged bodies are surveyed
by Wehausen (1973) and in other references listed in section 6.13.
282
Chapter 6
6.10 Three-Dimensional Ship Waves
Two-dimensional results can be generalized to more realistic threedimensional motions if the single plane wave system in the wake, shown
in figure 6.12, is replaced by a suitable distribution of waves moving at
all possible oblique angles θ. The most general wave distribution in three
dimensions is given by the double integral (59). If this is transformed
to a reference system moving with the ship in the positive x-direction
with velocity U, the appropriate expression with x replaced by x + Ut is
given by
∞
2π
0
0
η( x, z , t ) = Re ∫ dω ∫ dθ A(ω ,θ ) exp[ −ik( x cos θ + z sin θ ) + i(ω − kU cos θ )t ]. (86)
Here k(ω) is the wavenumber corresponding to a given frequency ω in
accordance with the dispersion relation (17) for infinite depth or (24) for
finite depth. As in section 6.9, we shall treat only the former possibility.
If the motion is steady state in the reference system which moves with
the ship, the expression (86) must be independent of time. As in the corresponding two-dimensional case, this implies a restriction on the phase
velocity of the waves and thus on the wavenumber. In the present situation, (86) will be independent of time provided
kU cos θ − ω = 0;
(87)
this is equivalent to stipulating that the phase velocity of each admissible
wave component is given by
Vp =
ω
= U cos θ .
k
(88)
In this sense (see figure 6.6), a system of plane progressive waves moving at
an oblique angle θ with respect to the x-axis will appear steady state to an
observer moving along the x-axis with velocity Vp secθ.
The restriction (87) can be used to eliminate one of the variables of integration in (86). Retaining the wave angle θ, and noting that (87–88) require
that cosθ > 0, we can replace (86) by the single integral
η( x, z ) = Re
π /2
∫
− π /2
dθ A(θ )exp[ −ik(θ )( x cos θ + z sin θ )];
(89)
Waves and Wave Effects
283
for the deep-water case, from (17) and (87),
k(θ ) = g / U 2 cos2 θ .
(90)
The corresponding expression for finite depth follows from (24), in combination with (87). In either case, the velocity potential corresponding to (89)
is of the same general form and can be inferred from (16) or (23).
The expression (89) is known as the free-wave distribution of a given ship.
This is characterized by the amplitude and phase of the complex function
A(θ); here A(θ) takes a role analogous to that of the constant A in section
6.9. Thus energy arguments can be used to express the wave resistance in
terms of the function A(θ), which depends on the shape of the ship hull
and its forward velocity. This reduction will be deferred to section 6.12.
If the distance downstream from the position of the ship is very large,
the integral (89) can be simplified and the classical ship-wave pattern will
be obtained, as derived by Kelvin in 1887. Strictly speaking this asymptotic
approximation should be based upon the stationary-phase approximation
of section 6.11. However, the Kelvin wave pattern is of sufficient interest
that a heuristic argument will be used in the interim, based on the concept
of the group velocity.
We recall from section 6.7 that in two dimensions wave groups travel
with the group velocity Vg = dω/dk. Since the free-wave distribution (89)
appears locally as a slowly varying two-dimensional wave system, the
same result should hold here. Thus if x′ is a local coordinate, normal to
the wave crests as in figure 6.6 and defined with respect to a fixed reference frame, the significant waves will travel in groups such that x′/t = Vg is
determined by
d
( kx′ − ωt ) = 0.
dk
(91)
If this expression for the derivative of the phase of the local waves is transformed into the frame of reference of (89), the significant waves will satisfy
the equation
d
[ k(θ )( x cos θ + z sin θ )] = 0.
dk
(92)
Since k and θ are related by (90), the vanishing of (92) is equivalent to
the condition
284
d  x cos θ + z sin θ 
= 0,

cos2 θ
dθ 
Chapter 6
(93)
except for the isolated points θ = 0, ±π/2 where dθ/dk is infinite or zero,
respectively. Ignoring these singular points and evaluating the derivatives
in (93) yields
x sec 2 θ sin θ + z sec 3 θ (1 + sin2 θ ) = 0.
(94)
Thus, the significant waves, moving in a direction θ, will be situated along
the radial line
z
cos θ sin θ
=−
.
1 + sin2 θ
x
(95)
A plot of (95) is shown in figure 6.14, and it is apparent that the maximum value of the ratio z/x is defined by
z / x = 2 −3/2 tan(19°28′ ),
(96)
this maximum occurs when the wave angle is
θ = ± sin −1(1 / 3 ) = ±35°16′.
Figure 6.14
Plot of equation (95).
(97)
Waves and Wave Effects
285
Thus, the waves are confined to a sector symmetrical about the negative
x-axis, with included semiangles of 19 21 °. Within this sector, for a given
point (x, z) and corresponding ratio z/x, two solutions of (95) exist; hence
two waves with distinct angles θ move in different directions. By requiring
that the phase function in parentheses in (93) be a constant, we may calculate the loci of the crests as shown in figure 6.15.
The waves on the negative x-axis move in the same direction as the ship,
with θ = 0 as in the two-dimensional case. As the lateral coordinate z is
increased, the angle of these transverse waves changes, in accordance with
(95) and figure 6.14, reaching a value of ±35° on the boundaries. Larger values of θ correspond to the diverging waves; these have shorter wavelength
and converge toward the origin as shown in figure 6.15. On the boundaries
where these two wave systems meet with a common angle of ±35°, they differ in phase by a quarter wavelength. This particular detail will be explained
from the method of stationary phase.
The wave systems generated by real ships are shown in two aerial photographs, figures 6.16 and 6.17. In both cases, the qualitative features of
the Kelvin wave pattern are confirmed. Figure 6.17, taken directly above
the wake of the ship, permits a quantitative confirmation of the two angles
noted, although in this case the transverse waves are not obvious. Figure
6.17 also shows a commonly observed feature, namely that the apex of the
sectors containing the Kelvin waves is displaced upstream from the ship’s
bow by an amount typically as large as one ship length.
Figure 6.15
The Kelvin ship-wave pattern.
286
Chapter 6
Figure 6.16
Kelvin ship waves as observed in an aerial photograph. The transverse waves can be
seen in this photograph, although the steeper diverging waves appear to be dominant. Note the nodal lines which divide the diverging waves along two or three radial
lines, with 180° phase shifts across these lines. (Courtesy of U. S. Navy)
A departure from the linearized potential theory that appears in figures
6.16 and 6.17 is a pronounced breaking wave, originating at the bow and
trailing aft on either side of the ship into the wake. The effect of this breaking wave on the resistance of the ship, a particularly interesting topic in
contemporary naval architecture, will be discussed in section 6.14.
6.11 The Method of Stationary Phase
A more mathematical approach to the Kelvin ship-wave system can be
based upon the method of stationary phase, developed originally by Kelvin
Waves and Wave Effects
287
Figure 6.17
Aerial photograph of ship waves with the angles 19 21 ° and 35° superposed. A single
nodal line can be detected on each side by the 180° phase shift as in figure 6.16.
(From Newman 1970)
for the same purpose. This will furnish an alternative viewpoint to the heuristic argument based on the group velocity and will provide the necessary
information regarding the wave amplitude for subsequent use in analyzing
the energy flux and wave resistance.
An asymptotic approximation is desired for the integral (89), based on
the assumption that the distance from the origin is very large compared to
the wavelength. To derive the stationary-phase approximation in a more
general context, let us consider the integral
I = ∫ F(θ )e iRG(θ ) dθ ,
(98)
where F(θ) and G(θ) are arbitrary regular functions and R is a large parameter such as the polar radius. The function F may be complex, but G will
be assumed real. For large values of R, the phase RG will vary rapidly, and
the integrand of (98) will be a highly-oscillatory function. The resulting
288
Chapter 6
oscillations of the integrand will tend to cancel out over the range of integration, except locally at points of stationary phase where the derivative
G′(θ) vanishes.
Thus the principal contribution(s) to the integral (98) will come from
the point(s) of stationary phase, assuming that such points exist in the
range of integration. To evaluate the contribution to (98) from the vicinity
of such a point, say at θ = θ0, the function G(θ) may be expanded in a Taylor
series of the form
G(θ ) = G(θ 0 ) + 21 (θ − θ 0 )2 G ′′(θ 0 ) + .
(99)
Here G″ is the second derivative, and the term involving the first derivative
has been deleted since, by definition,
G ′(θ 0 ) = 0.
(100)
Substituting (99) in the integral (98), the contribution from the vicinity of
this point will be given by
θ0+ε
I
∫
dθ F(θ )exp{iR[ G(θ 0 ) + 21 (θ − θ 0 )2 G ′′(θ 0 )]}.
(101)
00 − ε
Here ε is a small quantity whose value depends on the variation of F and G
but not on R.
The integral (101) may be simplified by substituting the new variable of
integration
ξ = (θ − θ 0 )( 21 RG ′′(θ 0 ) )1/2 .
(102)
It follows that
I
( 21
F(θ 0 )
1/2
RG ′′ )
(
)
1/2
ε 21 RG ′′
(
∫
− ε 21 RG ′′
dξ exp[ i( RG ± ξ 2 )],
(103)
)
1/2
where the function F has been approximated by its value at θ0, and the
± sign is the same as the sign of the second derivative G″(θ0). In the limit
R →∞, the last integral can be evaluated using the definite integrals
∞
sin
∫ cos ξ dξ = (π / 2)
−∞
2
1/ 2
.
(104)
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289
Substituting (104) in (103) gives the desired asymptotic approximation
of (99) in the form
 2π 
I F(θ 0 ) 
 RG ′′ 
1/2
exp[ i( RG ± π / 4)].
(105)
This expression is valid, for sufficiently large values of R, unless G″ = 0.
If there are multiple points of stationary phase in the domain of integration, the contribution from each may be treated separately, according to
(105), and added together to give the total contribution from all relevant
points. The singular case G″ = 0 occurs if two points of stationary phase
coalesce, and here it is necessary to retrace the steps from the Taylor series
(99) with the addition of a term involving the third derivative. Further
details are given by Lamb (1932) and Stoker (1957).
Having derived the stationary-phase approximation (105), we return to
the ship-wave integral (89). Cartesian coordinates (x, z) are retained, with
the understanding that the results are valid for sufficiently large values
of the polar radius R = (x2 + z2)1/2. The relevant phase function is
RG(θ ) = ( g / U 2 )( x sec θ + z sec 2 θ sin θ ),
(106)
and the points of stationary phase are given by (93). Thus the method of
stationary phase confirms the analysis of the ship-wave pattern that was
based on the group velocity in section 6.10.
We can now obtain more information regarding the asymptotic form of
the free-wave distribution (89), including the amplitude of the waves; from
(105) the waves take the form
 2π 
η = Re A(θ ) 
 RG ′′ 
1/2
exp{i[ RG(θ ) ± π /4 ]}.
(107)
This expression is to be summed over both points of stationary phase
defined by (93) and figure 6.14. The second derivative G″ can be computed,
and its sign is opposite that of the slope of the curve plotted in figure 6.14.
Thus, the plus or minus sign in the phase of (107) corresponds respectively
to the transverse and diverging wave systems; a net phase shift of 90° occurs
between these as shown in figure 6.15.
The amplitude of the waves in (107) is proportional to R−1/2. This rate of
attenuation could be anticipated since in three dimensions the energy flux
(73–74) spreads out radially in the horizontal plane.
290
Chapter 6
These results can be extended to ship waves in finite depth, starting with
(89) and the appropriate relation from (23) for k(θ). For subcritical Froude
numbers, U/(gh)1/2 < 1, the wave pattern is similar to the deepwater situation, but with a larger included angle which ultimately approaches 90° as
U/(gh)1/2 → 1. At supercritical Froude numbers, transverse waves are ruled
out, since these are unable to move with sufficient phase velocity to maintain a steady position. The diverging waves that remain are confined within
an included angle similar to the Mach cone in supersonic aerodynamics.
The details are worked out in an early paper by Havelock (1963); some corrections appear in Kinoshita (1943).
6.12 Energy Radiation and Wave Resistance
The wave resistance of a three-dimensional ship can be related to the
energy radiation downstream in the wake, as in the corresponding analysis
of the two-dimensional problem. A control volume is considered, bounded
downstream by a vertical control surface x = constant, which moves with
the ship as in figure 6.12(b). The energy of the fluid in this control volume
is constant, and the energy flux across the control surface can be equated to
the work done to overcome the wave resistance.
In a fixed reference frame, the energy of each plane-wave component
moves in the direction θ with the group velocity Vg. The velocity of energy
transfer across the control surface, which moves through the fluid with
velocity U, is given by Vg cosθ − U. Multiplying by the energy density 21 ρ gA2
and integrating along the width of the control surface, we find that the
total energy flux moving in the positive x-direction is
∞
dE 1
= ρ g ∫ A2 (Vg cos θ − U ) dz.
dt 2 −∞
(108)
This energy-flux integral can be extended to the case where the amplitude A and direction θ are functions of the position (x, z), if the changes
in these quantities are small over distances comparable to a wavelength.
The diverging and transverse components of the Kelvin ship-wave system are slowly varying in this sense, provided the distance downstream
from the ship is large compared to the wavelength. With this caveat, (108)
can be used to determine the energy flux of the diverging and transverse
waves.
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291
Since the distance downstream is large, the stationary-phase approximation (107) can be used to determine the wave amplitude in the form
A = A(θ ) (2π / RG ′′ )1/2 .
(109)
With (63) and (88), the group velocity is given by
Vg = 21 Vp = 21 U cos θ .
(110)
Combining these results and setting (108) equal to the negative of the work
DU as in (81) yields
∞
D = πρ g ∫
−∞
A(θ )
2
RG ′′(θ )
(
1−
1
2
cos θ
2
) dz.
(111)
For each lateral position z, the two corresponding wave angles θ of the Kelvin wave system must be determined from (95), and the contribution from
each component must be included in (111).
The coordinate z can be eliminated in (111) by transforming the variable
of integration from z to the wave angle θ. Using the relation (94) with x
held fixed, we can derive dz/dθ from
x
d
d
dz
(sec2 θ sin θ ) + z [sec 3 θ (1 + sin2 θ )] = − [sec 3 θ (1 + sin2 θ )].
dθ
dθ
dθ
(112)
On the other hand, (106) is to be evaluated with the position (x, z) fixed,
and this gives the relation
{
RG ′′(θ ) = ( g / U 2 ) x
}
d
d
[sec 2 θ sin θ ] + z [sec 3 θ (1 + sin2 θ )] .
dθ
dθ
(113)
Equations (112) and (113) are combined, so that the derivative relating the
two variables of integration can be written as
dz
RG ′′
U 2 cos3 θ RG ′′
=
=
.
2
3
2
dθ ( g / U )sec θ (1 + sin θ )
g (2 − cos2 θ )
(114)
Using the last result, we can recast (111) in the relatively simple form3
D=
π /2
1
πρU 2 ∫ A(θ ) 2 cos3 θ dθ .
2
− π /2
(115)
Equation (115) expresses the wave resistance of a moving vessel as the
weighted integral of the square of the wave amplitude. The factor cos3θ
292
Chapter 6
implies that the dominant portion of the resistance will be associated with
the transverse waves.
An alternative derivation of (115) can be based on the energy flux passing through a pair of longitudinal planes z = constant on both sides of the
ship. The integrand of (108) is then replaced by A2 (Vgsinθ)dx. An approach
independent of the stationary-phase approximation results from substitution of the free-wave potential corresponding to (89) in the energy-flux
integral (74). The derivation for a transverse plane x = constant is outlined
by Havelock (1963, pp. 391–393).
As a result of the positive-definite nature of·(115), the effects of interference are limited, and one cannot expect to find dramatic results of the type
illustrated in the two-dimensional case by figure 6.13. Waves generated at
different locations along the ship may still interfere, but in a manner that
depends on θ and on the point of observation in the wake. By judicious
design or chance, the wave amplitude may vanish at discrete positions in
the wake. This is illustrated in figure 6.16 where the nodal values of the
diverging wave system are indicated by the 180° phase shift along a few
radial lines. Additional nodes may be expected for the shorter wavelengths
as θ → 90°, but these are obscured and of little significance to (115). In
any event, these nodes occur only for discrete values of θ. In general, A(θ)
will be nonzero, contributing to a positive and nonvanishing value of the
drag (115).
To utilize (115), the wave amplitude function A(θ) must be predicted
from theory or measured in a suitable experiment. These possibilities are
described separately in sections 6.13 and 6.14.
6.13 Thin-Ship Theory of Wave Resistance
The thin-ship theory of wave resistance was introduced by J.H. Michell in
1898 as a purely analytic approach for predicting the wave resistance of
ships. The essential assumption is that the hull is thin, that is, the beam is
small compared to all other characteristic lengths of the problem. The resulting solution can be expressed in terms of a distribution of sources on the
centerplane of the hull, with the local source strength proportional to the
longitudinal slope of the hull. This solution is analogous to the thickness
problem of thin-wing theory, as discussed in chapter 5. However, the shipwave problem is complicated by the need to choose the source potential, in
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293
the form suggested by equation (4.78), to satisfy the free-surface boundary
condition.
The necessary source potential that satisfies the free-surface boundary
condition is a complicated function; its derivation may be found in Wehausen and Laitone (1960). We content ourselves here by noting that this potential must possess an asymptotic approximation far downstream of the same
form as the Kelvin wave system (107), with the velocity potential locally a
plane wave of wavenumber specified by (90). In addition, this source potential must depend in a suitably symmetrical4 manner on the position of
the source and of the field point where the potential is measured.
If the resulting expression for the source potential is integrated over the
centerplane z = 0 of the ship, the wave amplitude can be expressed in the
form
A(θ ) =
2
∂ζ
( g /U 2 )sec 3 θ ∫∫
exp[( g /U 2 )sec 2 θ ( y − ix cos θ )]dx dy.
π
∂x
(116)
Here z = ± ζ(x, y) defines the local half-beam of the hull surface.
While (116) is given here without derivation, its form is not unreasonable. The source strength is proportional to ∂ζ/∂x, as in the thickness problem of lifting-surface theory, and for the reasons noted the far-field form
of the source potential is proportional to the exponential factor shown in
(116). Therefore the only part of (116) that cannot be guessed judiciously is
the multiplicative factor outside the integral!
If the amplitude function (116) is substituted in (115), the result is a
particular form of Michell’s integral
D=
4ρ g 2
πU 2
π /2
∂ζ
∫ sec θ ∫∫ ∂x exp[( g /U
3
2
2
)sec 2 θ ( y − ix cos θ )]dx dy dθ .
(117)
0
This multiple integral is not the sort of expression one expects to find in
a table of integrals, particularly since in practical cases the longitudinal
slope of the ship hull cannot be expressed in terms of simple mathematical functions. Nevertheless, a fairly large number of numerical computations have been carried out, based on this expression or other equivalent
forms, both for practical ship geometries and for simplified mathematical
bodies.
To determine the practical value of this theoretical approach to wave
resistance, one may compare numerical computations based on Michell’s
294
Chapter 6
integral with measurements from towing-tank experiments. However, this
type of comparison is subject to the uncertainties of Froude’s hypothesis
(equation 2.24). Typical results are reproduced in figure 6.18 for a destroyer
form, which must be classified as being closer to a thin ship than would
be the case for a supertanker. At higher Froude numbers the agreement is
reasonable, particularly in the context of experimental variations resulting
from different turbulence stimulators and model lengths. At lower speeds,
however, the theory appears to exaggerate the effects of interference by
comparison with the experiments.
A common explanation for this discrepancy is that viscous effects suppress the interference effects in a real fluid. An alternative possibility results
from the fundamental assumption of thin-ship theory that the beam is
small compared to all other length scales. The wavelength is significant
in this context and should be recognized as an additional length scale not
present in the steady-state lifting-surface problem. Since λ ∝ U2, it is inevitable that the Michell approximation will break down as the Froude number becomes small. Theories which seek to overcome this limitation are
discussed by Gadd (1976) and by Noblesse and Dagan (1976).
The Michell theory has been extended and applied to a broad range of
problems in ship hydrodynamics. Several relevant papers and comprehensive bibliographies may be found in the proceedings edited by Inui and
Kajitani (Society of Naval Architects of Japan, 1976), as well as in Wehausen
(1973). The contributions to this field by Havelock (1963) are particularly
notable, and a monograph by Kostyukov (1959) is devoted to the theory of
ship waves and wave resistance.
6.14 Wave Pattern Analysis
The use of the wave-resistance integral (115) with experimental measurements of the amplitude function A(θ) to determine the total wave resistance
provides a direct measurement of the wave resistance without recourse to
Froude’s hypothesis. This approach is known as wave pattern analysis. A significant feature of this technique is that the measured quantity A(θ) is linear in the ship’s disturbance, as compared to the quadratic wave resistance.
By relating changes in the wave amplitude to changes in hull form, a linear
optimization of the hull shape can be achieved more directly than is possible by studying the effects of shape on the total wave resistance.
Waves and Wave Effects
295
Figure 6.18
Comparison of the wave resistance coefficient calculated from Michell’s integral
(117) with the residual drag coefficient as measured from model tests. (From Graff,
Kracht, and Weinblum 1964; reproduced by permission of the Society of Naval
Architects and Marine Engineers)
The wave pattern analysis requires a relatively complex survey of the
wake region to determine the amplitude function A(θ) for all relevant wave
angles. By comparison, Froude’s approach only requires a simple measurement of the total drag force on the model. Furthermore, the Froude approach
accounts for the possibility of other contributions to the total resistance,
such as the viscous form drag. Unless these additional contributions are
296
Chapter 6
known or can be neglected, the wave resistance component as given by
(115) is of limited interest from the standpoint of predicting the total drag
on a full-scale ship.
On the other hand, the wave-pattern analysis is a valuable diagnostic
tool, particularly when used with a measurement of the momentum defect
in the viscous-wake region. These techniques have led to the discovery of
an additional drag component associated with wave breaking, and to a better understanding of the bulbous bows of supertankers and other ships with
low Froude number.
In fine high-speed vessels such as destroyers and passenger liners, a bulbous bow promotes beneficial interference between the waves generated at
different points along the length of the hull. Thus, for such vessels, the bow
bulb reduces the wave resistance. Originally, bulbous bows of a similar form
were fitted to supertankers on the basis of experimental measurements indicating significant reductions in the total drag, but these reductions often
exceed the total estimated wave resistance.
This apparent paradox has been reconciled by careful experimental measurements of the wave-energy flux and of the momentum defect in the
wake due to the viscous form drag. The latter measurement has revealed
the existence of momentum associated with the breaking waves that are
apparent in figures 6.16 and 6.17. Thus the wave-breaking resistance results
from the breaking of waves near the ship, predominantly at the bow. The
energy lost in this manner is convected downstream in the form of largescale turbulence or eddies.
For supertankers and similar vessels, the bulbous bow is effective in
reducing the magnitude of the bow wave and thereby in avoiding wave
breaking. For predicting the total drag, one may argue that it makes little
difference whether the energy is wasted in wave radiation or in wave breaking, but only through an understanding of the mechanisms involved can
the total drag be reduced intelligently and systematically.
An extensive discussion of wave-pattern analysis is given by Eggers,
Sharma, and Ward (1967). More recent references on this topic and on
wave-breaking resistance are contained in the symposium volume issued
by the Society of Naval Architects of Japan (1976).
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297
6.15 Body Response in Regular Waves
A subject of great interest to ocean engineers and naval architects is the
effect suffered by a floating or submerged vessel in the presence of ocean
waves. The types of bodies of importance here include fixed structures and
freely floating vessels, as well as the intermediate category of moored vessels. The most common responses of concern are the oscillatory motions of
a free body or the structural loading imposed upon a fixed body.
In the simplest case it may be assumed that the waves incident upon the
body are plane progressive waves of small amplitude, with sinusoidal time
dependence. The results we will obtain about bodies in regular waves are
not without physical interest, but their practical value might be questioned
given the highly irregular nature of actual waves in the ocean. Fortunately,
these two topics are connected by the description of irregular waves as a
linear superposition of sinusoidal components. The actual ocean environment is most realistically described as a random process, which causes a
random response of the body. These random motions will be described
in sections 6.20 and 6.21, after we have dealt first with the problem of
regular waves.
Plane progressive waves were analyzed in sections 6.2 and 6.3 on the
assumption that the wave amplitude is sufficiently small to justify linearization. The same assumption is made here, not only to be consistent with
the description of the incident waves, but also to permit superposition of
different facets of the wave-body interaction in regular waves and ultimately to justify the superposition of these to represent the more realistic
motions in irregular waves.
In the problem to be treated here, plane progressive waves of amplitude
A and direction θ are incident upon the body, which moves in response to
these waves with six degrees of freedom as illustrated in figure 6.19. Following the indicial notation used in chapter 4, we define three translational
motions parallel to (x, y, z) = (x1, x2, x3) as surge, heave, and sway, and three
rotational motions about the same axes as roll, yaw, and pitch, respectively.
The corresponding velocities Uj(t) will be sinusoidal in time, with the same
frequency as the incident waves, and thus
U j (t ) = Re(iωξ j e iωt ),
j = 1, 2,… , 6.
(118)
298
Chapter 6
Figure 6.19
Definition sketch of body motions in six degrees of freedom.
Here the complex amplitude is employed in preference to the body
velocity, since the hydrostatic restoring forces that will be encountered
are proportional to the amplitudes of heave, roll, and pitch. These restoring forces are analogous to the familiar “spring constant” of a mechanical
oscillator.
If the incident wave is sufficiently small in amplitude and the body is
stable, the resulting motions will be proportionally small. The velocity
potential ϕ can then be written in the form


 6
φ( x, y , z , t ) = Re  ∑ ξ jφ j ( x, y , z ) + Aφ A ( x, y , z ) e iωt  .

 j =1

(119)
This decomposition of the total potential may be compared with equation
(4.101), for the motions of a rigid body in an infinite unbounded fluid.
The most important difference here is the presence of an additional term
due to the incident waves. A more subtle difference is that in chapter 4 the
coordinates (xi) were fixed with respect to the body, and velocity potentials
ϕj for each mode of motion were expressed without approximation in terms
of that moving reference frame. Here, because of the free surface, we adopt
a fixed coordinate system. Under the assumption that (119) is a small firstorder quantity, the distinction between the inertial coordinate system and
one fixed in the body will be a source of second-order effects that can be
neglected.
Waves and Wave Effects
299
In (119), the function ϕj represents the velocity potential of a rigidbody motion with unit amplitude, in the absence of the incident waves.
For example, if the body is forced by an external mechanism to oscillate
in heave with unit amplitude, in otherwise calm water, the resulting fluid
motion will be represented by the potential ϕ2. A similar interpretation
applies for each mode of rigid-body motion. The appropriate boundary
conditions to be imposed on the body surface SB are obtained by equating
the normal derivative of (119) to the normal component of the body velocity, as in equations (4.102–4.103). Since the amplitudes ξj are independent
parameters, it follows that
∂φ j
= iω nj ,
∂n
j = 1, 2, 3,
∂φ j
= iω ( r × n ) j −3 ,
∂n
j = 4, 5, 6,
(120)
(121)
on SB. Here n is the unit normal vector on the body surface, directed into
the body, and r is the position vector (x, y, z). The forced motion potentials ϕj, j = 1, 2, …, 6, are generally known as solutions of the radiation
problem.
The remaining potential represented by the last term in (l19) is due to
the incident waves and their interaction with the body. Given the assumption of linear superposition, this potential is independent of the body
motions and may be defined with the body fixed in position. The appropriate boundary condition on the body surface is
∂φ A
= 0, on
∂n
SB .
(122)
The problem so defined is known as a wave diffraction problem.
The presence of the body in the fluid results in diffraction of the incident wave system and the addition of a disturbance to the incident wave
potential associated with the scattering effect of the body. This process can
be emphasized by the additional decomposition
φ A = φ0 + φ7 ,
(123)
where ϕ0 is the incident wave potential and ϕ7 is the scattering potential
that must be introduced to represent the disturbance of the incident waves
by the fixed body. The incident wave potential may be regarded as known,
300
Chapter 6
from (16) for infinite depth or from (23) for finite depth, and the scattering
potential must satisfy the boundary condition
∂φ7 / ∂n = −∂φ0 / ∂n, on SB .
(124)
In addition to the body boundary conditions (120–124), each potential
ϕj must satisfy Laplace’s equation
∇ 2φ j = 0 ,
j = 0,1,… ,7,
(125)
throughout the domain of the fluid, and the condition (21) on the bottom
for finite depth, or alternatively ϕj → 0 as y → −∞ for infinite depth. On the
free surface, the linearized boundary condition follows from (6) and (119)
in the form
−
∂φ j
ω2
φj +
= 0, on
∂y
g
y = 0,
j = 0,1,… ,7.
(126)
At this stage, boundary conditions have been prescribed on the body
surface, free surface, and bottom; but the boundary-value problem is not
unique. In particular, an arbitrary constant times the diffraction potential
(123) can be added to any one of the potentials ϕj, j = 1, 2, …, 7, without
violating these boundary conditions. To overcome this problem it is necessary to impose a radiation condition at infinity, which states that the waves
on the free surface, other than those due to the incident potential itself, are
due to the presence of the body. Thus, the waves associated with the potentials ϕj, j = 1, 2, …, 7, must be radiating away from the body. The appropriate
condition in two dimensions is
φ j ∝ e ∓ ikx , as x → ± ∞,
j = 1, 2,… ,7.
(127)
In three dimensions, with energy conservation or the stationary-phase
approximation as a guide, the waves at infinity must be of the general form
φ j ∝ R−1/2 e − ikR , as R → ∞,
j = 1, 2,… ,7.
(128)
Here R = (x2 + z2)1/2, and the constants of proportionality in (127) and (128)
may depend on the remaining coordinates, but not on x or R, respectively.
With the time dependence (119), these are the most general forms of outgoing waves due to the presence of the body. The incident wave potential ϕ0
is, by definition, excluded from these radiation conditions.
Waves and Wave Effects
301
While the radiation conditions may appear contrived or arbitrary, the
need for them results from our assumption of sinusoidal time dependence
for all previous time. A similar situation was noted in sections 3.13 and
5.16, in dealing with viscous diffusion and unsteady lifting surfaces. In
those cases appropriate initial conditions were prescribed, to make the solutions unique. The same approach could be used here with the final results
unchanged, but imposing the radiation conditions (127–128) is simpler.
This subject is discussed at some length by Stoker (1957).
The general form of the oscillatory force and moment acting on the
body can be inferred by substituting (119) in Bernoulli’s equation. Retaining only the first-order linear terms in (3), the total pressure is given by
∂φ
p = −ρ 
+ gy 
 ∂t



 6
= − ρ Re  ∑ ξ jφ j + A(φ0 + φ7 ) iω e iωt  − ρ gy.


 j =1

(129)
The force F and moment M can be determined by integrating the fluid pressure (129) over the wetted surface SB. Substituting (129) in (4.81–4.82) gives
expressions of the form
6
 F
 n 
 n 
iωt
 M = − ρ g ∫∫  r × n y dS − ρ Re ∑ iωξ j e ∫∫  r × n φ j dS
j =1
SB
SB
 n 
− ρ Re iω Ae ∫∫ 
(φ0 + φ7 ) dS.
 r × n
SB
(130)
iωt
The three integrals in (130) represent distinctly different contributions
to the total force and moment. The first is the hydrostatic component,
which is both elementary and important. The second integral in (130) is
reminiscent of the added-mass coefficients studied in chapter 4, except that
in the present case the potentials ϕj are complex. The resulting force and
moment are defined generally in terms of the damping and added-mass coefficients, corresponding to the real and imaginary parts of these integrals.
Finally, the last term in (130) is the exciting force or moment, proportional
to the incident wave amplitude. Each of these three contributions will be
discussed separately, in sections 6.16–6.18.
302
Chapter 6
6.16 Hydrostatics
Hydrostatics is the oldest and most elementary topic of naval architecture
and fluid mechanics. Our task here is to evaluate the first term in (130), for
the hydrostatic force
F = − ρ g ∫∫ ny dS ,
(131)
SB
and moment
M = − ρ g ∫∫ ( r × n )y dS ,
(132)
SB
taking advantage of the linearization assumption. The coordinate y is fixed
in space, whereas the wetted surface SB oscillates with the body, and for a
floating body the upper boundary of SB is the free surface y = η(x, z, t), as
shown in figure 6.20.
Since the hydrostatic pressure is of order one, the distinction between
the fixed coordinates (x) and body coordinates (x′) is significant. Assuming
the body motions ξj are small and neglecting the second-order quantities,
we can express the transformation between these coordinate systems as
Figure 6.20
Two-dimensional sketch showing the fixed and moving coordinate system, wetted
surface SB of the vessel, and free surface y = η.
Waves and Wave Effects
303
(133)
x = x′ + xT + x R × x′.
Here xT = (ξ1, ξ2, ξ3) and x R = (ξ4, ξ5, ξ6) are vectors formed by the translation
and rotation of the body; the latter are unique since the motions are small.
Throughout this section, ξj will be used to denote the real time-dependent
body motion. In the context of hydrostatics, ξj may be a constant or an
arbitrary function of time.
The dominant portions of (131) and (132) are the static buoyancy force
and moment, which are canceled for an unrestrained body by its weight.
Since the center of gravity moves with the body, it is convenient to replace
(132) by the moment M′ about the origin of the body-fixed coordinates x′.
Thus
M′ = − ρ g ∫∫ [( r − xT ) × n ]y dS.
(134)
SB
Before evaluating the force (131) and moment (134), we note that
the integrands are proportional to y; therefore the contribution from the
thin strip 0 < y < η will be of order η2. This second-order quantity may be
neglected in the linearized analysis, and the surface of integration SB in
these integrals can be redefined as the instantaneous body surface beneath
the calm-water plane y = 0.
To evaluate these surface integrals, the surface SB may be closed by adding to it the interior portion of the plane y = 0, where the integrand is zero.
The volume (t ) within this closed surface is the instantaneous displaced
volume of the body beneath the plane y = 0. Applying Gauss’s theorem to
(131) and the corresponding theorem (4.92) to (134) yields
F = ρ g ∫∫∫ (∇y )d  = jρ g ∫∫∫ d  ,

(135)

M′ = − ρ g ∫∫∫ [ ∇ × y( r − xT )]d 

= ρ g ∫∫∫ [ k( x − ξ1 ) − i( z − ξ3 )]d  .
(136)

The force (135) is obviously the buoyancy force proportional to the displaced volume (t ), and the moment (136) is the cross-product of this
buoyancy force with the vector position of the center of buoyancy.
The volume integrals in (135–136) can be evaluated in terms of the bodyfixed coordinates, after decomposing  into the static volume 0 beneath
304
Chapter 6
the plane y′ = 0 and the thin layer bounded by the planes y = 0 and y′ = 0.
From (133) it follows that


F = jρ g  ∀ − ∫∫ (ξ2 + ξ6 x′ − ξ4 z ′ ) dx′ dz ′ ,


S0
(137)

M′ = ρ g ∫∫∫ [ k( x′ + ξ5z ′ − ξ6 y ′ ) − i( z ′ + ξ4 y ′ − ξ5 x′ )]d 
 0

− ∫∫ (ξ2 + ξ6 x′ − ξ4 z ′ )( kx′ − iz ′ )dx′ dz ′  ,
S0

(138)
where 0 and S0 denote the displaced volume and waterplane in the static
condition.
Equations (137) and (138) can be expressed in the form
F = jρ g[ ∀ − ξ2 S + ξ4 S3 − ξ6 S1 ],
M′ = − iρ g[ ∀( z B + ξ4 y B − ξ5 xB ) − ξ2 S3 + ξ4 S33 − ξ6 S13 ]
+ kρ g[ ∀( xB + ξ5z B − ξ6 y B ) − ξ2 S1 + ξ4 S13 − ξ6 S11 ].
(139)
(140)
Here ∀ is the displaced volume, S the waterplane area, xB the center of
buoyancy
xB =
1
x d ,
∀ ∫∫∫
0
(141)
and the waterplane moments are defined as
Sj = ∫∫ xj dS ,
j = 1, 3,
(142)
S0
Sij = ∫∫ xi xj dS.
(143)
S0
Note that the integrals (141–143) are defined in terms of the static condition
of the vessel, and the primes have been deleted with this understanding.
The horizontal components of the hydrostatic force and the vertical
component of the moment are identically zero. For a submerged body the
waterplane area S and moments (142–143) vanish, and (139–140) simplify
accordingly.
In the conventional terminology of naval architecture, (S1/S, 0, S3/S) are
the coordinates of the center of flotation. This is the point of rotation for a
Waves and Wave Effects
305
freely floating body subject to an applied horizontal moment. The integrals
(142) vanish if the origin is placed at the center of flotation, and with a
suitable rotation of the horizontal coordinates it can be assumed that S13 =
0 as well.5 This particular choice of the coordinate system will be assumed
hereafter. Thus the vertical force (139) is independent of the roll and pitch
motions, and the terms in (139–140), which involve S1, S3, and S13 can be
deleted.
For subsequent use it is convenient to add to the hydrostatic force and
moment the corresponding components due to the weight of the body.
The force associated with the body weight is simply (0, −mg, 0), where m
is the body mass. The corresponding moment is given, to first order in the
motions ξj, by the vector
( mgzG + mgyGξ4 − mgxGξ5 , 0, − mgxG − mgzGξ5 + mgyGξ6 ).
Using an indicial notation for the components of the resulting total static
force (F1, F2, F3), and moment (F4, F5, F6), it follows from (139–140) that
6
Fi = ( ρ∀ − m ) gδ i 2 + ( mzG − ρ∀z B ) gδ i 4 − ( mxG − ρ∀xB ) gδ i 6 − ∑ cijξ j .
(144)
j =1
Here δij is the Kroenecker delta function, equal to one if i = j and zero otherwise, and cij is the matrix with nonzero elements
c22
c44
c45
c65
c66
= ρ gS ,
= ρ gS33 + ρ g ∀y B − mgyG ,
= − g ( ρ∀xB − mxG ),
= − g ( ρ∀z B − mzG ),
= ρ gS11 + ρ g ∀y B − mgyG .
(145)
For other values of (i, j), cij = 0.
The expression (144), for the total static force and moment, consists of
terms of zero- and first-order in the body motions ξj, with second-order
terms neglected. If the body is in equilibrium, for ξj = 0, the zero-order
terms must vanish. For equilibrium of the vertical force, the mass of a freely
floating body must equal the displaced mass,
m = ρ∀,
(146)
in accordance with Archimedes’ principle. With this equality, equilibrium
of the zero-order moments in (144) requires that the center of gravity and
center of buoyancy must lie on the same vertical line,
306
Chapter 6
xB = xG ,
(147)
z B = zG .
(148)
For a freely floating body, where (146–148) hold, the elements of the
matrix coefficients (145) contained in parenthesis vanish, leaving as the
only nonzero coefficients
c22 = ρ gS ,
c44 = ρ g ∀[( S33 / ∀ ) + y B − yG ],
c66 = ρ g ∀[( S11 / ∀ ) + y B − yG ].
(149)
The body is statically stable if the first-order components in (144) oppose
small displacements ξj. Since there is no static restoring force or moment in
surge, sway, and yaw, these are neutrally stable modes. In heave, roll, and
pitch of a freely floating body, the necessary and sufficient condition for
static stability is that the corresponding coefficients of (149) are positive.
In heave this condition is always satisfied, provided the waterplane area is
nonzero. A submerged body, with S = 0, is neutrally stable in heave, or neutrally buoyant, provided the condition (146) is satisfied.
Since static stability in roll and pitch requires that the coefficients c44
and c66 be positive, it follows from (149) that stability in these modes will
depend on the height of the center of gravity. The factors in square brackets in these expressions are the vertical distances between the metacenters
( Sjj / ∀ ) + y B and the center of gravity yG. These differences in elevation
are known as metacentric heights and often are denoted by the symbol
GM. The metacenters of a submerged body coincide with the center of
buoyancy, but for a floating body the metacenters are above yB, since the
moments of inertia Sjj about the center of flotation are positive. Thus the
center of gravity of a stable floating body may be situated above the center
of buoyancy.
Comprehensive treatments of static stability for ships are given by Moore
(1967) and by Rawson and Tupper (1968).
6.17 Damping and Added Mass
In chapter 4 we showed that if a body moves in an infinite ideal fluid,
hydrodynamic pressure forces and moments will result which can be
expressed most simply in terms of the added-mass coefficients mij. A more
Waves and Wave Effects
307
complicated and less general situation follows for a body moving on or near
a free surface; the corresponding force and moment are expressed by the
second term on the right side of (130), with the restrictions that the body
motions are small and sinusoidal in time.
The six components of this force and moment can be written in the
matrix form

 6
Fi = Re ∑ ξ j e iωt f ij  , i = 1, 2,… , 6,
 j =1

(150)
where
f ij = − ρ ∫∫
SB
∂φ i
φ j dS.
∂n
(151)
The last result follows from (130) and the boundary conditions (120–121).
The coefficient fij is the complex force in the direction i, due to a sinusoidal
motion of unit amplitude in the direction j.
The coefficient defined by (151) is analogous to the added-mass coefficient mij defined by (4.114) for a body undergoing small oscillations in
an unbounded fluid. Here, however, we shall find that the coefficients fij
are complex as a result of the free surface, and the real and imaginary parts
depend on the frequency ω. For this reason the coefficients (151) take the
form
f ij = ω 2 aij − iω bij .
(152)
Equivalently, the force (150) can be expressed in the form
6
Fi = − ∑ ( aijU j + bijU j ),
(153)
j =1
which is a decomposition of the sinusoidal force, associated with each
mode of motion, into components in phase with the velocity and acceleration of the corresponding modes.
The coefficient aij is known as the added-mass coefficient, since it represents the force component proportional to the acceleration. It should be
emphasized that aij will differ from the corresponding added-mass coefficient mij for a body in the absence of a free surface, and one must not
assume that all of the properties of mij apply to aij.
308
Chapter 6
The coefficient bij in (153) gives a force proportional to the body velocity; for this reason bij is known as the damping coefficient. The presence of
such a force results from the generation of waves on the free surface, due to
the motions of the body. These waves radiate outward, with a corresponding energy flux that can be computed from (73–78). As in the steady-state
wave-resistance problem, work must be done to oscillate the body and generate the radiating waves. For a single mode of body motion, the average
work done to oppose the pressure force (153) over one cycle is
2
2
2
1
− FU
i i = bii U i = 2 ω bii ξi .
(154)
Thus, there is a direct relation between the damping coefficients and the
amplitude of the waves generated by the body. For any body motion where
waves are generated on the free surface, the damping coefficient must be
greater than zero, and therefore the force coefficient fij must be complex.
One simplification that the damping and added-mass coefficients share
with the unbounded-fluid case is symmetry. The proof follows in an analogous manner to (4.117–4.118); however, Green’s theorem must be applied
not only on SB but to the complete closed surface, including the free surface, bottom, and a control surface at infinity. Fortunately, a consequence
of the boundary conditions on each of these additional surfaces is that
none of them contribute to the integrand of (4.117). The two terms in this
integrand vanish separately on the bottom and cancel on the free surface
when (126) is imposed. The same cancellation occurs on the control surface
at infinity if the two potentials satisfy the radiation conditions (127–128).
Thus (4.117) applies here, in the form

∫∫  φ
SB
i
∂φ j
∂φ 
− φ j i  dS = 0, i , j = 1, 2,… , 6;
∂n
∂n 
(155)
from (151) it follows that
f ji = f ij , i , j = 1, 2,… , 6.
(156)
Since the same result holds separately for the real and imaginary parts of fij,
the damping and added-mass coefficients are symmetric.
The dependence of aij and bij on frequency can be verified by considering
the two limits ω → 0 and ω → ∞. In these cases the free-surface condition
(126) takes the form
Waves and Wave Effects
∂φ j / ∂y = 0
on y = 0, ω → 0,
309
(157)
and
φj = 0
on y = 0, ω → ∞.
(158)
These can be regarded as the limits where gravitational forces dominate
inertial forces, and vice versa. (Strictly, since ω and g are dimensional quantities, one should state that (157) and (158) are valid respectively for the
two limiting values of the nondimensional ratio ω2l/g, where l is the characteristic body length.)
Equation (157) is the rigid-wall boundary condition, where the free
surface is replaced by a fixed horizontal plane. The resulting problem for
each ϕj can be solved by the method of images, as outlined in section 4.18.
The appropriate image above the plane y = 0 is a body of the same geometrical form, reflected about y = 0. The phase of the image body motion
must be such that the potential is an even function of y, and the normal
velocity takes the same value at corresponding points on the body and its
image.
For low-frequency motions in the horizontal plane (surge, sway, and
yaw), the image body must move with the same phase as the real body, as
shown in figure 6.21. In this case the body plus its image form a rigid double
body, and the potentials ϕj, j = 1, 3, 5, can be related directly to the corresponding solutions of chapter 4 for a rigid body in an unbounded fluid. The
limiting values of the corresponding added-mass coefficients are
aij → 21 mij ,
(159)
for i, j = 1, 3, 5, as ω → 0, with mij the double-body added mass.
For low-frequency motions in the vertical plane (heave, roll, or pitch),
symmetry requires the phase of the image body to be opposite that of the
real body, as shown in figure 6.21. In this case the body plus its image can
Figure 6.21
The low-frequency limit ω → 0 for surge, sway, or yaw (left), and for heave, roll, or
pitch (right).
310
Chapter 6
still be regarded as a double body in an unbounded flow, but the mode of
motion corresponds to an oscillatory dilation of the body. The potentials ϕj
must be derived separately, and the resulting added-mass coefficients aij are
not related to the corresponding values of mij. As a particular example, the
solution for heave will require a net oscillatory source strength to match
the changing volume of the double body, with a completely different farfield behavior relative to the dipole-like potentials (4.123 and 4.129).
The opposite situation applies in the high-frequency limit (158), where
the potential must be odd in y, and the resulting asymmetry requires the
image phase to be as shown in figure 6.22. Here, the vertical modes of
motion correspond to those of a rigid double body, and thus (159) will apply
for i, j = 2, 4, 6, as ω → ∞. On the other hand, the horizontal modes correspond to shearing deflections of the double body that cannot be related to
the potentials obtained in chapter 4 or to the resulting values of mij.
Comparing figures 6.21 and 6.22 shows that the added-mass coefficients
aij must differ for ω → 0 and ω → ∞, since different types of double-body
motions result in these two limits. It follows that the added-mass coefficients must depend on ω, in order to pass from one limit to the other as
0 < ω < ∞.
The damping coefficients vanish at the two limits, since the respective
double-body flows are in an unbounded fluid without waves. Since the
damping coefficients are greater than zero for intermediate frequencies,
these too must depend on ω.
Many investigations have been carried out to compute the damping and
added-mass coefficients of particular body forms, as functions of frequency.
The majority of this work pertains to two-dimensional cylinders in deep
water. Several references and examples are given by Wehausen (1971).
For the two-dimensional case, Vugts (1968) has made extensive calculations; some of these are reproduced by Wehausen (1971) and in a revised
form in figure 6.23. In three dimensions systematic calculations have been
Figure 6.22
The high-frequency limit ω → ∞ for surge, sway, or yaw (left), and heave, roll, or
pitch (right).
Waves and Wave Effects
311
Figure 6.23
Added-mass and damping coefficients for a family of two-dimensional rectangular
cylinders, heaving in deep water, based on the computations of Vugts (1968). Included below is the thin-ship approximation given by (179) and labeled B/T = 0. Also
shown here by the scale on the right is the heave exciting-force coefficient, obtained
in accordance with (174).
312
Chapter 6
carried out by Kim (1966) for a series of ellipsoids, including the special
case of a floating hemisphere. The latter was first considered by Havelock
(1963), and the damping and added-mass coefficient for heave are reproduced in figure 6.24.
Since the complex force coefficient fij is an analytic function of ω in the
lower half-plane, the damping and added-mass coefficients can be related
by integral transforms known as the Kramers-Kronig relations. Further
details may be found in the surveys of Ogilvie (1964) and Wehausen (1971).
In addition, relations exist between the damping coefficients and exciting
forces; these will be discussed in the next section and utilized to derive the
damping coefficients for a thin ship.
Figure 6.24
Added-mass and damping coefficients for a sphere of diameter d, half submerged in
deep water. ∀ is the displaced volume πd3/12. Also shown is the heave-response ratio
(190).
Waves and Wave Effects
313
6.18 Wave-Exciting Force and Moment
The last term in (130) gives the exciting force and moment proportional to
the incident-wave amplitude,
Fex 
 n 
iωt
dS.
 = − ρ Re iω Ae ∫∫ (φ0 + φ7 ) 
 r × n
M ex 
SB
(160)
It will be convenient to define the six components of (160) in the form
Fexi = Re{ Ae iωt Xi }, i = 1, 2,… , 6.
(161)
Thus Xi is the complex amplitude of the exciting force or moment, in the
direction i, for an incident wave of unit amplitude. Using (160) and the
boundary conditions (120–121) it follows that
Xi = − ρ ∫∫ (φ0 + φ7 )
SB
∂φ i
dS.
∂n
(162)
For a prescribed incident wave, the potential ϕ0 can be found from (16)
or (23) for infinite or finite depth, respectively. The diffraction potential
ϕ7 must be determined as the solution of the boundary-value problem
(124–128).
Since the wave amplitude A is a first-order quantity, the surface integrals
in (160–162) can be evaluated on the mean position of the body surface,
with the neglected factor a second-order quantity proportional to A2. Moreover, none of the boundary conditions governing the potentials ϕ0 or ϕ7
is affected by the body motions. Thus, the linearized exciting force and
moment are independent of the body motions, just as the damping and
added mass are independent of the incident-wave amplitude. In particular,
(160–162) can be used to find the exciting force either for a fixed body or
for one free to respond to the waves. The distinction between these is a
second-order effect.
Since the boundary-value problem (124–128) differs from that for the
potentials ϕj, j = 1, 2, …, 6, only with regard to the boundary condition
(124) on the body surface, the diffraction potential ϕ7 can be considered
the disturbance associated with a forced normal velocity on the body surface equal and opposite to that of the incident wave. This suggests the
long-wave approximation where, if the body is small compared to the wavelength, the induced velocity field from the incident wave can be assumed
314
Chapter 6
constant over the body. Under this restriction the diffraction potential can
be approximated in the form
φ7 i  ∂φ 0
∂φ
∂φ

φ1 + 0 φ2 + 0 φ3  ,
ω  ∂x
∂y
∂z 
(163)
where the derivatives of ϕ0 are constants in this expression, evaluated at a
suitable position on or within the body.
Using the long-wave approximation (163), the contribution to (162)
from the diffraction potential ϕ7 can be evaluated in terms of the coefficients fij defined by (151),
− ρ ∫∫ φ7
SB
∂φ i
i 3
∂φ
dS = ∑ f ij 0 .
∂n
ω j =1 ∂xj
(164)
The integral in (162) involving ϕ0 can be evaluated from Gauss’s theorem.
Consistent with (163) and (164), the potential ϕ0 can be approximated by a
Taylor series, and it follows that
∫∫ φ
0
SB
n dS = − ∫∫∫ ∇φ0 d  − ∫∫ φ0 n dS
S0
0
∂φ
∂φ
−∀∇φ0 + j  φ0 S + 0 S1 + 0 S3  .

∂x
∂z 
(165)
The corresponding integral for the moment follows from (4.92),
∫∫ φ (r × n)dS = ∫∫∫ (∇ × φ r )d − ∫∫ φ (r × n)dS
0
0
SB
0
0
S0
∂φ
∂φ
∀∇φ0 × x B − i  φ0 S3 + 0 S13 + 0 S33 


∂x
∂z
∂φ
∂φ
+ k  φ0 S1 + 0 S11 + 0 S13 


∂x
∂z
(166)
where the hydrostatic parameters are defined in section 6.16. We shall
avoid the step of combining (165) or (166) with (164) and substituting in
(162), but we note that the derivatives of the incident wave potential will
be proportional to kϕ0. Since λ is large, the wavenumber k is small, and the
leading-order force and moment will result from the terms in (165–166)
proportional to ϕ0. Substituting these in (160), and using (5) yields
Fex jρ gSη(t ),
(167)
M ex ( kS1 − iS3 )ρ g η(t ).
(168)
Waves and Wave Effects
315
The leading-order contributions (167–168) are the hydrostatic force and
moment associated with the elevation of the incident wave at the body.
The remaining contributions from (164–166) will be of order (kl), or
proportional to l/λ, which is assumed small. The contribution from (164)
can be related to the damping and added-mass coefficients, and the remaining contributions from (165–166) can be computed from the hydrostatic
coefficients. A simple example is the vertical exciting force on a body symmetrical about the planes x = 0 and z = 0; this force can be obtained in
the form
X2 [ ρ gS + iω b22 − ω 2 ( a22 + ρ∀ )].
(169)
Here the leading term is hydrostatic, and the relative motion hypothesis is
suggested by the fact that the damping and added-mass coefficients in
(169) multiply the velocity and acceleration of the incident wave. The term
involving the displaced volume ∀ can be interpreted in a similar manner as
the corresponding term in (4.150), for the force on a body in an unbounded
fluid with a slowly-varying inhomogeneous velocity field. Equation (4.150)
can be used to compute the exciting force on a submerged body, under the
same assumptions as (169). The only difference that will result, apart from
the absence of the waterplane area S, is that the complex force −ω2a22 + iωb22
is replaced in (4.150) by −ω2m22.
For λ/l = O(1), the long-wave approximation is not valid, but the boundary conditions for ϕ7 and the forced-motion potentials ϕj, j = 1,2, …, 6, are
similar except on the body surface. We shall take advantage of this similarity to derive the Haskind relations, which express the damping coefficients
in terms of the exciting forces. No assumptions are required regarding the
wavelength or body geometry.
Since ϕ7 satisfies the same boundary conditions as ϕj on the free surface and bottom and the same radiation condition at infinity, Green’s
theorem in the form (155) can be applied to the diffraction potential. In
particular,

∫∫  φ
SB
i
∂φ7
∂φ
− φ7 i  dS = 0, i = 1, 2,… , 6.
∂n
∂n 
(170)
Substituting this result in (162), gives
∂φ
∂φ
Xi = − ρ ∫∫  φ0 i + φi 7  dS.
 ∂n
∂n 
SB
(171)
316
Chapter 6
The diffraction potential can be eliminated from (171) by substituting the
boundary condition (124), and the Haskind relations follow in the form
∂φ
∂φ
Xi = − ρ ∫∫  φ0 i − φi 0  dS.
 ∂n
∂n 
SB
(172)
The most significant feature of (172) is that the exciting force has been
expressed, without approximation, in a form independent of the diffraction potential ϕ7. Moreover, it follows from Green’s theorem that the integral (172) can be evaluated on a control surface at infinity. This integral
does not vanish, since the incident wave does not satisfy the radiation condition, but at infinity the forced-motion potential ϕi is proportional to the
square root of the energy flux, associated in turn with the damping coefficient bii.
In three dimensions the method of stationary phase (105) can be used
to show that
bii =
k
8πρ gVg
2π
∫
Xi (θ ) 2 dθ .
(173)
0
Here Xi(θ) is the amplitude of the exciting force, for waves of direction θ
and unit amplitude. Equation (173) is particularly useful for axisymmetric
bodies, with a vertical axis, where the heave exciting force is independent
of the wave direction and the remaining forces and moments are proportional to sin θ or cos θ. In this case the integral (173) can be evaluated, and
a simple algebraic relation follows (see problem 12).
Similarly, in two dimensions, for a body symmetric about the plane
x = 0, it can be shown that
bii =
Xi 2
,
2 ρ gVg
(174)
where Xi is the amplitude of the exciting force per unit length. Derivations
and extensions of these relations are given by Wehausen (1971) and Newman (1976). Vugts (1968) has confirmed (174) from experimental measurements of the damping coefficients and exciting forces.
The character of the exciting force and moment as ω → ∞ or λ → 0 can
be inferred from a physical argument and confirmed from the Haskind relations. Physically, if the wavelength is short compared to the body length,
several waves will exist simultaneously along the body and cancellation
Waves and Wave Effects
317
will occur. Moreover, the exponential decay of (16) with depth will result
in a significant effect of the incident waves only in a thin “boundary-layer”
region near the free surface. Alternatively, since the damping coefficients
vanish in the high-frequency limit, it follows from (173) that the threedimensional exciting force must tend to zero faster than (Vg/k)1/2, or using
(17) and (77),6 gives
Xi (θ ) = o(ω −3/2 ),
(175)
as ω → ∞. In two dimensions, it follows from (174) that
Xi = o(ω −1/2 ),
(176)
as ω → ∞. These estimates are conservative, and more precise high-frequency
limits for heave are given in the paper by Davis (1976), and references cited
therein. For sway, see problem 15.
The simplest representation for the exciting force and moment is based
on the assumption that the pressure field is not affected by the presence of
the body and can be determined from the incident wave potential by itself.
In other words, the diffraction potential ϕ7 is neglected completely in the
integrals (160–162). This approach was utilized in the earliest theories for
ship motions in waves; it is known as the Froude-Krylov hypothesis in honor
of the authors of those theories.
Historically, the Froude-Krylov approach was expedient, but in certain
cases it can be justified. For a thin ship, with centerplane z = 0, Peters
and Stoker (1957) have shown that the Froude-Krylov exciting force and
moment are the leading-order contributions in the vertical plane, i.e., for
surge, heave, and pitch (j = 1, 2, 6). A similar result is established for slender
bodies in chapter 7. Both proofs are valid only if the wavelength is large
compared to the beam.
To illustrate the Froude-Krylov approach for a thin ship, we start with
(162) and restrict our attention to motions in the vertical plane (i = 1, 2, 6).
The normal components nx, and ny can be approximated by the slopes
∂ζ/∂x and ∂ζ/∂y, respectively, where ζ(x, y) is the local half-beam which is
assumed small. From the boundary conditions (120–121), the normal derivatives ∂ϕi/∂n will be first-order quantities, proportional to the beam. For following or head waves (θ = 0 or 180°), it follows from the boundary condition
(124) that the diffraction potential ϕ7 also will be a small first-order quantity
proportional to nx and ny.
318
Chapter 6
For oblique waves, the normal component nz contributes to the boundary condition (124), and for this reason the effect of the diffraction potential on sway, roll, and yaw is not small. Nevertheless its contribution to
the vertical modes remains small in oblique waves, as a consequence of
symmetry.7
Since ϕ0 is a function of order one, independent of the beam of the ship,
the exciting force (172) is given to first order by
xi − ρ ∫∫ φ0
SB
∂φ i
dS , i = 1, 2, 6.
∂n
(177)
Restricting our attention to the heave-exciting force in deep water, we
obtain the potential ϕ0 from (16),and using ∂φ2 / ∂n iω∂ζ / ∂y as the thinship approximation of (120), we get
X2 2 ρ g ∫∫
∂ζ ky −ikx cosθ
e
dx dy.
∂y
(178)
Here the integral is over the centerplane z = 0, and the factor of two accounts
for both sides of the body. The surge-exciting force and pitch moment are
similar to (178), but with ∂ζ/∂x and (x ∂ζ/∂y − y ∂ζ/∂x) in place of ∂ζ/∂y.
These expressions are analogous to (116) for the waves generated by a thin
ship in steady forward motion.
The heave-exciting force acting on a thin two-dimensional cylinder can
be obtained as a special case of (178), where ζ depends only on y, and
the x-integration is from minus infinity to plus infinity. The exciting force
cancels along the length unless the incident wave crests are parallel to the
cylinder. In the latter case, the heave-exciting force, per unit length, is
given by
X2 2 ρ g
0
dζ ky
e dy ,
dy
−T
∫
(179)
where T is the draft, and the integral is over the vertical extent of the body.
Substituting the exciting force (178) in (173) gives the heave-damping
coefficient for a thin ship in the form
b22 =
ρ gk
2πVg
2π
∫
0
2
∂ζ
∫∫ ∂y e ky −ikx cosθ dx dy dθ .
(180)
Waves and Wave Effects
319
Similar expressions can be derived for surge and pitch, and the analogy
with Michell’s integral (117) should be noted.
For the two-dimensional cylinder, we use (174) and (179), to get
b22 =
2ρ g
Vg
0
2
∂ζ
∫ ∂y e ky dy .
−T
(181)
From this result one can deduce the wavemaker theory, for the waves generated by a vertical wavemaker oscillating horizontally at one end of a wave
tank (see problem 16). The wavemaker theory, as well as the damping coefficients (180) and (181), can be derived directly from solutions of the corresponding forced-motion potentials ϕi, as noted in Wehausen and Laitone
(1960). However, the Haskind relations (172–174) can be used to deduce
these results much more simply.
The proof of the Haskind relations is independent of the water depth,
and (178–181) can be generalized to the case of finite depth simply by
replacing the exponential factor eky by the ratio of hyperbolic cosines
in (22).
6.19 Motion of Floating Bodies in Regular Waves
In sections 6.16–6.18 the various components of the total pressure force
and moment (130) have been discussed, as well as the force and moment
due to the weight of the body. We are now in a position to derive the equations of motion for free oscillations of the body in waves by equating the
above forces to the inertial forces associated with acceleration of the body
mass. Here we assume that the body is rigid, unrestrained, and in a state of
stable equilibrium when in calm water.
The six components of the inertial force, associated with the body mass,
can be obtained from section 4.16. With the assumption of linearized
motions it follows that
6
Fi = ∑ M ijU j , i = 1, 2,… , 6,
j =1
where the matrix Mij is defined by
(182)
320
Chapter 6
 m
 0

 0
M ij = 
 0
 0

− myG
0
0
m
0
0
m
0 myG
0
0
0
0
0
0
myG
I11
I 21
I 31
0
0
0
I12
I 22
I 32
− myG 
0 

0 
.
I13 
I 23 

I 33 
(183)
Here the body mass is
m = ∫∫∫ ρ B d  ,
(184)
B
the mass-density is ρB, and the moments of inertia can be defined in the
form
I ij = ∫∫∫ ρ B[ x ⋅ xδ ij − xi xj ] d  , i , j = 1, 2, 3,
(185)
B
where δij is the Kroenecker delta function, equal to one if i = j and zero
otherwise.
Equations of motion follow by equating the inertia forces (182) to the
sum of the pressure forces (130) and the forces due to the body weight,
which are incorporated in the total static restoring forces (144). Combining
these relations, six simultaneous equations of motion are obtained in the
form
6
∑ ξ ( −c
j
6
ij
j =1
+ f ij ) + AXi = −ω 2 ∑ M ijξ j .
(186)
j =1
Rearranging this equation and replacing fii by the added-mass and damping
coefficients from (152), we arrive at the result
6
∑ ξ [ −ω
j
2
( M ij + aij ) + iω bij + cij ] = AXi .
(187)
j =1
These are six simultaneous linear equations, which can be solved for the
body motions ξj by standard matrix-inversion techniques. Thus, in general,
the body motion ξj will be given by an equation of the form
6
ξ j = A∑ [ Cij ]−1 Xi .
i =1
(188)
Waves and Wave Effects
321
where Cij denotes the total matrix in square brackets on the left side of
(187). The ratio (ξj/A) is a quantity of fundamental significance, which we
shall define separately by
6
Z j (ω ,θ ) ≡ ξ j /A = ∑ [ Cij ]−1 Xi .
(189)
i =1
Physically, this is the complex amplitude of body motion in the jth mode,
in response to an incident wave of unit amplitude, frequency ω, and direction θ. This ratio is known as the transfer function, or the response amplitude
operator. The transfer function can be calculated from (189) if the addedmass, damping, exciting, and hydrostatic forces are known.
To illustrate the nature of the transfer function (189), we return to the
special case of heave for a body symmetrical about x = 0 and z = 0. From
symmetry, there are no cross-coupling effects, except for possible coupling
between surge and pitch or between sway and roll. Thus, in heave,
Z2 (ω ,θ ) =
X2
X2
=
.
C22 −ω 2 ( a22 + M 22 ) + iω b22 + c22
(190)
In the limit of low frequencies and long wavelengths, the exciting force in
the numerator can be approximated by (169); then the limit is
Z2 (ω ,θ ) → 1,
as ω → 0.
(191)
Physically, if the waves are sufficiently long, the body will simply ride up
and down with the free surface.
At very high frequencies the denominator of (190) will be proportional
to ω2, while the exciting force in the numerator will tend to zero. Thus, as
ω →∞,
Z2 (ω ,θ ) = o(ω −7/2 ,ω −5/2 ),
(192)
for three and two dimensions, respectively. More precise estimates can be
developed from Davis (1976), and for horizontal motions as noted in problem 15.
Returning to the transfer function (190) for intermediate frequencies,
the most significant feature will be a resonant response at the natural frequency where the virtual mass and restoring forces cancel, or where
c22


ωn = 
 a22 + M 22 
1/2
.
(193)
322
Chapter 6
At or near this resonant frequency, the body will experience a response of
large amplitude and a phase shift increasing from zero for low frequencies
to 180 degrees for ω ≫ ωn.
As in the case of a mechanical oscillator, the resonant response will be
inversely proportional to the damping coefficient, but here there is a connection between the exciting force and the damping coefficient from the
Haskind relations. In particular, for two-dimensional or axisymmetric bodies the damping coefficient will be proportional to the square of the exciting force, and the resonant response will be inversely proportional to the
exciting force. Thus, bodies deliberately designed with small exciting forces
will experience a large resonant response, in a highly-tuned manner. Vertical spar buoys are important examples of this situation, as shown in figure
2.16 and in problem 17. By comparison, the floating hemisphere shown in
figure 6.24 is typical of the situation where the exciting force, damping, and
resonant response assume relatively moderate values.
The analogy with a mechanical oscillator can be emphasized by rewriting (187) in the time-dependent form
( M ij + aij )ξj (t ) + bijξ j (t ) + cijξ j (t ) = Re( AXi e iωt ).
(194)
However, the simplicity of this result is deceptive, and the analogy with
a mechanical oscillator must not be pushed too far. Equation (194) has
been derived assuming the motions are sinusoidal in time, as evidenced
by the fact that the added-mass and damping coefficients depend on the
frequency ω. Thus, one cannot extend (194) directly to the case where the
time-dependence is more general, although one can do so by suitable superposition of the incident waves and body motions at each frequency. We
shall return to this representation in section 6.21.
The distinction between the left side of (194) and a mechanical oscillator
with constant coefficients can be further emphasized by noting that when
the body is forced to oscillate, waves will be generated on the free surface.
As time increases, these waves will propagate outward from the body, but
they will continue to affect the fluid pressure and hence the body force for
all subsequent times. Thus memory effects are introduced, in a manner precisely analogous to the unsteady lifting surface theory in chapter 5.
One consequence of the memory effects is that if ξj(t) is an arbitrary
function of time, and if the linearization assumption is invoked, the pressure force acting on the body must be of the general form
Waves and Wave Effects
6
Fi (t ) = ∑
323
t
∫ K (t − τ )ξ (τ )dτ .
ij
j
(195)
j =1 −∞
This is a convolution integral, over the previous history of the fluid motion,
and the kernel K(t − τ) can be interpreted as the force, at time t, due to a
delta-function body velocity at an earlier time τ. This impulse-response function is discussed by Price and Bishop (1974). The analog in chapter 5 is the
Wagner function shown in figure 5.35.
The restriction of (194) to sinusoidal motions and the significance of the
convolution integral (195) have been emphasized by a number of authors
including Haskind (1953), Cummins (1962), and Maskell and Ursell (1970).
Maskell and Ursell point out that if a floating body in deep water is given
an impulsive motion at time t = 0 and is unrestrained thereafter, the body
will oscillate about its equilibrium position for a finite number of cycles,
approaching equilibrium asymptotically as an inverse power of time. By
comparison, the homogeneous solution of the differential equation (194),
with constant coefficients, would decay exponentially with an infinite
number of oscillations.
Throughout this discussion we have assumed that the body motions are
oscillatory about a fixed mean position. An important exception to this
situation is that of a ship, moving with forward velocity U while oscillating in response to ambient waves. The assumption of linearity permits this
oscillatory problem to be superposed on the steady-state wave-resistance
solution. Nevertheless there are fundamental effects of the forward speed
on the oscillatory problem, the simplest and most important being the
change in frequency due to the Doppler shift. The complete solution of
this problem is not established to the same extent as the simpler problem
with zero forward speed, but a strip-theory approach will be discussed in
chapter 7.
6.20 Ocean Waves
The complexity of ambient wave motions in the ocean is evident to anyone who has observed them either from the beach or the deck of a ship.
The wave patterns are ever-changing with time and space, in a manner
that appears to defy analysis. This complexity is partly the result of dispersion, which already has been discussed in the context of deterministic wave
324
Chapter 6
motions generated by well-defined processes. Ship waves are an example
where the uniformity of the resulting wave system is itself the most striking
feature of all.
Ambient waves on the surface of the ocean are not only dispersive, but
random. The generating mechanism is, predominantly, the effect upon the
water surface of wind in the atmosphere. The wind is itself random, especially when viewed from the standpoint of the turbulent fluctuations and
eddies which are important in generating waves. The randomness of ocean
waves is subsequently enhanced by their propagation over large distances
in space and time and their exposure to the random nonuniformities of
the water and air. Thus ocean waves must be described in a probabilistic
manner.
Despite their randomness, ocean waves can be modeled by a suitable
distribution of plane progressive waves of all possible frequencies and
directions, as in the double integral (59) where the wave motion is defined
uniquely by the complex amplitude A(ω, θ). In a deterministic problem
with finite energy, this amplitude function can be found by Fourier techniques, for example, in terms of prescribed values of the wave elevation
along the free surface, at some initial instant of time.
For random ocean waves this approach must be modified. First, the
required initial data cannot be measured everywhere on the free surface.
In addition, Fourier theory requires the function to be represented, that is,
η(x, z, t), to be square-integrable over the entire domain, in this case the free
surface. Ambient ocean waves do not vanish anywhere on the free surface,
and while their features may be noticeably different at widely separated
points in the ocean, an integral such as
∞
∫∫−∞ η
2
dx dz will be finite only in
the sense that the area of the oceans is limited.
Faced with a dilemma of whether the area of the oceans is finite or infinite, we find it appropriate to consider the length-scale and time-scale of
interest to us in our description of the ocean-wave environment. In the context of ocean engineering and naval architecture, our interest is to describe
the response of vessels and structures with dimensions comparable to a few
hundred meters, at the most, and with characteristic response times on the
order of a few seconds or, in extreme cases, minutes.
For this purpose, it should suffice to describe the wave environment
over a small portion of the oceans, say on the order of tens or hundreds of
kilometers squared. We have no interest in predicting simultaneously the
Waves and Wave Effects
325
environment throughout the world, or even throughout the North Atlantic Ocean. The complexity of developing such an extensive description,
assuming it to be possible, would outweigh completely the advantages of
making the classical Fourier theory valid!
By the same argument, we are interested in typical wave environments
that might exist in conjunction with storms of varying severity, but not
with the transient development of these waves with time. Therefore it
should suffice to describe the wave environment over a period of a few
hours, and to assume that the wave motion is stationary during this interval
of time. It corresponds to this assumption that the waves are homogeneous
in space over the area in question. These statements have meaning only in
the statistical sense, since for a random wave system it would be ridiculous
to suggest that the precise wave motion is the same at different points in
space or time.
Another consideration from the practical standpoint, which avoids the
need or desire for a deterministic solution, is that our interest in the wave
environment is confined not to any one instant of time, but to a typical
situation that might be expected over a representative period on the order
of several hundred wave encounters. For this reason, the precise phase of
any one wave is of no interest, and it is reasonable to assume that the phase
of each component is distributed randomly with equal probability between
0 and 2π.
This last assumption overcomes the problem of square-integrability in
the Fourier integral theory, but strictly it is necessary to replace the distribution (59) by a Fourier-Stieltjes representation of the form
η( x, z , t ) = Re ∫∫ dA(ω ,θ )exp[ −ik( x cos θ + z sin θ ) + iωt ].
(196)
Here the wavenumber k is defined in terms of the frequency ω by the dispersion relation (17) or (24). The differential wave amplitude dA is a complex
quantity of random phase, with a magnitude proportional to the square
root of the total energy in a differential element of the frequency-waveangle space bounded by (ω, ω + dω) and (θ, θ + dθ). This type of representation, described by Yaglom (1962), can be circumvented by using infinite
series of discrete waves in place of the continuous distribution (196).
The average energy density can be computed by squaring (196) and taking its average value; it then takes the form
326
η2 =
Chapter 6
1
2
∫∫ dA(ω ,θ )exp[ −ik(ω )( x cos θ + z sin θ ) + iωt ]
(197)
× ∫∫ dA * (ω ′ ,θ ′ )exp[ ik(ω ′ )( x cos θ ′ + z sin θ ′ ) − iω ′t ].
Here (*) denotes the complex conjugate. The only contribution to this average is from the combinations (ω, θ) = (ω′, θ′), with the result
η2 =
∞ 2π
1
dA(ω ,θ ) dA*(ω ,θ ) ≡ ∫
2 ∫∫
0
∫ S(ω ,θ ) dθ dω.
(198)
0
Unlike the wave amplitude A, the function S(ω, θ) is regular and the last
integral can be interpreted in the ordinary Riemann sense. Derivations of
this result that avoid the subtleties of (196–197) can be found in Neumann
and Pierson (1966) and Price and Bishop (1974).
This approach to random ocean waves has been motivated by the deterministic distribution of plane waves and the desire to extend that representation to the random case. The spectral density S(ω, θ) is more commonly
defined probabilistically as the Fourier transform of the correlation function for the free-surface elevation. This description is outlined in Phillips
(1966) and in the statistical theory of turbulence by Monin and Yaglom
(1971).
Comparing (198) and (69), we can infer that the total mean energy of
the wave system per unit area of the free surface is equal to
∞ 2π
E = ρg ∫
∫ S(ω ,θ ) dθ dω.
(199)
0 0
It is customary to ignore the factor ρg and to refer to the function S(ω, θ) as
the spectral energy density, or simply the energy spectrum. More specifically,
this is a directional energy spectrum; it can be integrated over all wave directions to give the frequency spectrum
S(ω ) =
2π
∫ S(ω ,θ ) dθ.
(200)
0
If one attempts to find the ocean-wave spectrum from measurements of
the free-surface elevation at a single point in space, for instance by recording the heave motion of a buoy, the directional characteristic of the waves
will be lost. Only the frequency spectrum (200) can be determined from
such a restricted set of data. A limited amount of directional information
Waves and Wave Effects
327
follows if one measures the slope of the free surface, for example by measuring the angular response of the buoy as well as its heave. A complete
description of the directional wave spectrum requires an extensive array of
measurements at several adjacent points in space, however, and there are
practical difficulties associated with this task.
As a simpler alternative, one can assume that the waves are unidirectional, with the energy spectrum proportional to a delta function in θ.
Wave spectra of this form are called long crested, since the fluid motion is
two-dimensional and the wave crests are parallel. If one assumes that the
spectrum is of this character, with the direction prescribed, the frequency
spectrum (200) is sufficient to describe the wave environment.
If the waves are generated by a single storm, far removed from the point
of observation, it might be presumed that these waves would come from
the direction of the storm in a long-crested manner. The limitations of this
assumption are apparent to anyone who has observed the ocean surface.
While a preferred direction may exist, especially for long swell that has
traveled large distances, even these long waves will be distributed in their
direction, and for short steep waves the directional variation is particularly
significant. Since the superposition of such waves from a range of different
directions appears in space as a variation of the free-surface elevation in all
directions, these waves are known as short-crested waves.
Information on directional spectra is limited, although the knowledge
of this subject is likely to increase as sophisticated measurement techniques
and data analysis are applied. More extensive discussion of this subject is
given by Kinsman (1965), Neumann and Pierson (1966), Lewis (1967), and
Price and Bishop (1974).
In part from necessity, but with the presumption that a conservative
estimate of wave effects on vessels will result, it is customary to proceed in
the fields of ocean engineering and naval architecture on the assumption
that the waves are long-crested. With this simplification it is possible to use
existing information for the frequency spectrum (200), which is based on a
judicious combination of theory and full-scale observations.
Ocean-wave spectra depend on the velocity of the wind as well as its
duration in time and the distance over which the wind is acting on the free
surface. This distance is known as the fetch. Wave spectra that have reached
a steady-state of equilibrium, independent of the duration and fetch are
328
Chapter 6
said to be fully developed. A semi-empirical expression for the frequency
spectrum of fully developed waves is
S(ω ) =
ag 2
exp[ − β ( g /U ω )4 ].
ω5
(201)
Here α and β are nondimensional parameters defining the spectrum, and
U is the wind velocity at a standard height of 19.5 meters above the free surface. This two-parameter spectrum is sufficiently general to fit most observations and is consistent with theoretical predictions of the high-frequency
limit.
The most common values for the parameters in (201) are
α = 8.1 × 10 −3 ,
(202)
β = 0.74;
(203)
with these values, (201) is known as the Pierson-Moskowitz spectrum. The
resulting family of curves is shown in figure 6.25, for wind speeds of 10 to
50 knots.
If the spectrum is assumed to be of narrow bandwidth,8 the wave amplitudes follow a Rayleigh distribution. Thus the probability that the wave
amplitude A lies within the differential increment (A, A + dA) is p(A) dA,
where the normalized probability density function is given by
p(ζ ) = ζ e −ζ
2 /2
(204)
.
Here, ζ = A/m01/2 is the normalized wave amplitude, and m0 is the total
energy of the spectrum, or the integral of S(ω) over all frequencies. More
generally the moments mj of the spectrum are defined as
∞
mj = ∫ ω j S(ω ) dω ,
j = 0,1, 2,….
(205)
0
Statistics for the wave height H = 2A can be derived using the probability
density function (204). Thus the average wave height H is given by
∞
H = 2 ∫ Ap(ζ ) dζ = (2π m0 )1/2 .
(206)
0
For most purposes, however, we are interested primarily in the larger waves.
The most common parameter that takes this into account is the significant
Waves and Wave Effects
329
Figure 6.25
The Pierson-Moskowitz frequency spectrum (201–203) for the wind speeds in knots
as listed in table 6.1. The peak value for 50 knots is at S = 54 m2-s. The curve for 10
knots cannot be shown on this scale, its peak value being S = 0.02 m2-s.
wave height H1/3 , defined as the average of the highest one third of all the
waves. This can be computed as the ratio
∞
H1/3 = 2 ∫ Ap(ζ ) dζ
ζ0
∞
∫ p(ζ )dζ = 4.0(m
0
)1/2 ,
(207)
ζ0
where the lower limit of the integrals is defined such that
∞
∫ p(ζ ) dζ = 1/3.
(208)
ζ0
Averages that focus upon less frequent maxima can be defined in a corresponding manner; for example, the average of the highest one tenth of all
waves can be defined as
H1/10 = 5.1( m0 )1/2 .
(209)
330
Chapter 6
The average frequency of the spectrum can be defined as the expected number of zero upcrossings per unit time, that is, the number of times the wave
elevation passes through zero with positive slope. The derivation is given
by Price and Bishop (1974), and the final result is
∞
ω ≡  ∫ ω 2 S(ω ) dω
0
∞

∫ S(ω ) dω 
0
1/2
m 
=  2
 m0 
1/2
(210)
.
The moments m0 and m2 can be computed without difficulty for the twoparameter spectrum (201), and it follows that
H1/3 = (2.0 )
U2 α
g  β 
1/2
(211)
,
and
ω = (πβ )1/4 ( g /U ).
(212)
Substituting the values (202–203) for the Pierson-Moskowitz spectrum gives
the results shown in table 6.1. Also shown in this table is the approximate
Beaufort scale of the sea state for each wind speed and the average value
of the period, as derived by Neumann and Pierson (1966). The last column
shows the average wavelength corresponding to each wave period, based
on the dispersion relation for plane progressive waves.
The results shown in table 6.1 are based on the assumption of a fully
developed sea. It is unusual for a storm of 50 knots’ intensity to last long
enough for the sea to develop fully. Thus, while wind speeds of this intensity are not uncommon, the corresponding wave height and length shown
in table 6.1 are rare.
Table 6.1
Statistical Parameters and Approximate Sea State of the Pierson-Moskowitz Spectrum, for the Wind Speeds Shown in Figure 6.25
Significant
Wave Height
(meters)
Average
Period
(seconds)
Average
Wavelength
(meters)
Wind Speed
(knots)
Sea State
(approx.)
10
2
0.6
2.7
11
20
4
2.2
5.3
45
30
6
5.0
8.0
100
40
7
8.9
10.7
178
50
8
13.9
13.4
278
Waves and Wave Effects
331
Much of the uncertainty regarding the process by which the wind speed
is related to the waves can be avoided by identifying the Pierson Moskowitz spectrum in terms of the significant wave height rather than the wind
speed or sea state. The situation is somewhat more complicated than this,
however. If the sea is not fully developed, the corresponding wavelengths
will be shorter since dispersion has less opportunity to occur. Thus, the
Pierson-Moskowitz spectrum underestimates the peak frequency for the
higher spectra and conversely for smaller waves. To account for this problem, the two-parameter spectrum (201) is used with full-scale observations
of the significant wave height and period. The parameters α and β are then
determined from (211–212).
Fetch-limited wave spectra measured during the Joint North Sea Wave
Project (JONSWAP) show higher peaks compared to the Pierson-Moskowitz
spectrum and the two-parameter family (201). The engineering implications of the JONSWAP results are discussed by Houmb and Overik (1976).
The significant wave height is relevant as an indication of the most likely
waves to be encountered, but for design purposes a more conservative estimate is required. Thus we are led to consider the statistics of extreme values.
From the Rayleigh distribution (204), the cumulative probability of a
single wave being less than A = ζ(m0)1/2 is
ζ
P(ζ ) = ∫ p(ζ ′ ) dζ ′ = 1 − e −ζ
2 /2
.
(213)
0
The probability that the amplitudes of N statistically independent waves are
all less than A is PN, and the probability that at least one wave amplitude
will exceed this value is
2
1 − P N = 1 − (1 − e −ζ /2 )N .
(214)
The negative derivative of (214) gives the probability density of the extreme
value in N cycles, in the form
g (ζ ) = N p (ζ )[ P(ζ )]N −1.
(215)
The wave amplitude for which this probability density function is a
maximum can be found by setting the derivative of (215) equal to zero.
Simplifying the resulting expression for N ≫ 1, the most probable extreme
value in N waves is given by
332
Chapter 6
A = (2m0 log N )1/2 .
(216)
Ochi (1973) has shown that the probability of exceeding (216), for large
N, is 1 − e−1 = 0.632. Thus a more conservative criterion is required, and for
this purpose we define the design extreme value as the wave amplitude that
will be exceeded in N encounters with a probability of only 1 percent. From
(214) it follows that
1 − P N = 0.01,
(217)
and thus
P = ( 0.99)1/ N = e(1/ N )log( 0.99 ) e −0.01/ N 1 − 0.01 / N .
(218)
Combining the last result with (213) and solving for the wave amplitude,
gives
N 
A =  2m0 log

0.01
1/2
.
(219)
This is the design extreme value, or the maximum wave amplitude in N
encounters that will not be exceeded with probability 0.99. Note from
(216) that this result is equal to the most probable extreme value in 100N
encounters.
Since the significant wave amplitude is half the wave height (207), the
design extreme value exceeds the significant value by the ratio
Design Extreme  1
N  1/2
log
.
=
2
Significant
0.01
(220)
This factor is not sensitive to the value of N, nor to the arbitrary choice
of 0.01 as the probability that the extreme wave will exceed this value.
For example, with N = 100, (220) is equal to 2.15; for N = 104 this factor
increases to 2.63. Since (219) is much more sensitive to the value of m0 than
to (N/0.01), the most important factor to estimate correctly is the total wave
energy.
In a comprehensive derivation of the extreme-value statistics, Ochi
(1973) shows that the amplitude of the design extreme wave is reduced if
the wave spectrum is not of narrow bandwidth. However, the net reduction
is less than 10 percent in all cases.
Waves and Wave Effects
333
6.21 Motions of Bodies in Irregular Waves
With the spectral description of ocean waves given in the preceding section, we can return to the problem of body motions and generalize the
earlier treatment of regular waves with sinuosidal time-dependence. If
the ocean waves are described by the random distribution (196), and if the
response of the body to each component wave is defined by a linear transfer
function Z(ω, θ), the body response will be
ξ j (t ) = Re ∫∫ Z j (ω ,θ )e iωt dA(ω ,θ ).
(221)
The principal assumption is that linear superposition applies, as it must
in any event for the underlying development of the transfer function and
spectrum.
Like the waves themselves, the response (221) is a random variable. The
mean-square of this quantity can be computed in precisely the same manner as (197–198), giving the result
∞ 2π
ξ 2j = ∫
∫ S(ω ,θ ) Z (ω ,θ )
j
2
dω dθ .
(222)
0 0
The distribution of response maxima is approximated by a Rayleigh distribution. Thus the statistics of the body response are identical to the wave
statistics as analyzed in section 6.20, except that the wave-energy spectrum
S is multiplied by the square of the transfer function.
To a large extent these relations provide the justification for our earlier
analysis of regular-wave response. The transfer function Zj(ω, θ) is valid not
only in regular waves, where it has been derived, but also in superpositions
of regular waves, and ultimately in a spectrum of random waves. Generally
speaking, a vessel with favorable response characteristics in regular waves
will be good in irregular waves, and vice versa. This statement is oversimplified, however, and the relative “shape” of the energy spectrum and the
transfer function is crucial to the value of integrals such as (222). For example, a large resonant response of the body will be of importance if the resonant frequency coincides with the peak of the wave-energy spectrum, and
vice versa. Figure 2.16 shows the extreme resonant peak which results for
a slender spar buoy, but the spar buoy FLIP, described by Rudnick (1967),
is sufficiently large that the natural period in heave (27 seconds) is beyond
the range of expected wave excitation. Similarly, a small unrestrained buoy
334
Chapter 6
will have a high natural frequency in heave; as such its resonant response
will be of little significance. From the corresponding limits (192) and (191)
for the transfer function in heave, it is obvious that FLIP will be stable
with practically no heave motion, while the small buoy will follow the freesurface elevation. Resonance is not significant in either example for normal
conditions.
As in the case of the ambient waves, the phase of the response (221) is
a random variable of no importance. However the relative phase, between
the response ξ(t) and the wave η(t), or between two different modes of
response such as pitch and heave, may be a quantity of great significance.
For example, the relative motion (ξ2 − η) between the body and the free
surface determines the probability of submergence or emersion. Thus one
may be concerned whether these random variables are in phase or out of
phase. Here the argument of the transfer function is relevant, and can be
used to determine the correlation between the different random variables
noted above. For heave, we recall that for frequencies below the resonant
frequency the argument of the transfer function is close to zero, whereas
at high frequencies the body and waves move with a phase difference of
180°. For floating bodies, as for mechanical systems, coupling between two
modes such as heave and pitch will introduce a pair of natural frequencies,
both of which must be considered in this context.
Little has been said here about motions in the horizontal plane, where
no resonance exists. It is essentially for the latter reason that these modes
are of less importance, but their analysis can be carried through in much
the same manner as has been outlined here for heave. Resonant horizontal motions may result from an elastic mooring, and second-order drift
processes in the horizontal plane are of practical importance in some
cases. Both subjects are beyond the scope of this text, but references can
be found in Bishop and Price (1975). Additional references are noted in
section 2.15.
Problems
1. For plane progressive waves of height 6 m and length 200 m in deep
water, find the phase velocity, the maximum velocity of a fluid particle,
and the position where this maximum occurs. How do these change if the
fluid depth is 30 m?
Waves and Wave Effects
335
2. Give the velocity potential for standing waves in finite depth, which
reduces to (53) as kh → ∞. Show that this corresponds to the solution of a
problem where incident waves are totally reflected by a vertical wall. Using
the linearized Bernoulli equation, find the horizontal wave force exerted
on the wall. Compute the magnitude of this force if A =
1
2
m, h = 10 m, and
λ = 50 m.
3. The mass transport (49) can be computed to second order in finite depth
by taking the time average of the horizontal flux across a vertical control
surface, between −h and η, after noting that ϕx = −(k/ω)ϕt for a progressive wave and that the time average of ϕt must be zero if the motion is
periodic. In this manner show that the generalization of (49) in the form
Q = 21 gA2 / Vp is valid for finite depths.
4. Derive expressions for the kinetic and potential energies of a standing
wave in finite depth, and show that the total energy per wavelength is constant. Since the standing wave is equivalent to two identical plane progressive waves propagating in opposite directions, show that the energy of each
plane wave is divided equally into kinetic and potential portions.
5. Compute the maximum electric power that can be generated by a device
that converts all of the energy from plane progressive waves, of height
1 m and wavelength 100 m, over a width of 1 km parallel to the wave
crests.
6. A wavemaker at x = 0 is turned on at time t = 0. At t = 10 s the maximum
possible wave amplitude is desired at x = 20 m. Show that the appropriate
frequency for the wavemaker is ω = C1 − C2t, assuming deep water. Find the
constants C1 and C2. Describe qualitatively how this result can be changed
to account for finite-depth effects.
7. A ship model of length 5 m is moved with a constant speed of l m/s
for a distance of 100 m, in a wide towing tank. At the end of this run the
model is stopped, and thereafter it is affected by the transverse waves coming upon it from astern. What is the period of these waves, and what is their
wavelenth? Assuming the back edge of this wave system is sharp, separating
a region of calm water from the region of the transverse waves, how many
waves exist in the towing tank at the time the model is stopped? How many
waves will ultimately move past the model after it has stopped?
8. Waves of period five seconds are incident upon a gradually shoaling
beach in a two-dimensional manner. At what depth will the wave height
336
Chapter 6
be a minimum? If the wave height 2A in deep water is 1 m and if the waves
break when the depth is equal to the local wave height, what is the wave
height when breaking commences? Explain why regular waves incident
upon a beach from an oblique angle must appear steady-state to an observer
moving along the beach at a suitable velocity, for all depths. Show from this
that the waves are refracted toward the beach; that is, show that the wave
crests become more nearly parallel to the beach as the depth decreases.
9. Show that a round homogeneous log is neutrally stable, with respect to
static roll about its axis, for any density such that it will float. Show that
the position of stable equilibrium of a square homogeneous log depends
on its density.
l0. A rectangular barge of length 100 m, beam 20 m, and draft 8 m is assumed
to have an added mass equal to its displaced mass. Find the resonant wavelength for heave if the barge is moored with a slack mooring cable.
11. From energy conservation, show that the damping coefficient of a twodimensional body symmetrical about x = 0 is related to the amplitude Ai
of the far-field waves, generated by its motion in deep water, by the equation
2
bii =
ρ g 2 Ai
.
ω 3 ξi
Substitute this result in (172) and carry out the vertical integration at infinity to derive the two-dimensional version of the Haskind relations in the
form (174).
12. Using (173), show that the heave damping coefficient of a body of revolution with vertical axis is equal to
b22 =
k
X2 2 .
4ρ gVg
Show that for sway or surge a similar expression holds with a multiplicative
factor of one half.
13. For a two-dimensional floating body, show that a suitable linear combination of heave and sway oscillations at the same frequency will result
in outgoing waves only at x = ∞, with no waves at x = − ∞. By reversing the
sign of time t, deduce that the corresponding body motions act to absorb
an incident wave system with 100 percent efficiency.
Waves and Wave Effects
337
14. Using the long-wave approximation (169) in conjunction with the
exact result (173), obtain the first two nonvanishing terms in power-series
expansions for the damping and exciting force, in powers of the frequency,
for heaving motions in deep water. Compare these results with the exact
calculations for a floating hemisphere shown in figure 6.24.
15. Consider the horizontal exciting force on a floating body in the shortwavelength limit λ/l ≪ 1. If the body surface is vertical at the intersection with the free surface, explain the fact that the body can be replaced
by a vertical cylinder with the same waterplane and large depth. Use the
standing wave solution (53) to estimate the sway exciting force on a twodimensional body for λ/l ≪ 1.
16. Show from the respective boundary conditions that the wavemaker
problem, of a prescribed horizontal velocity along a vertical wavemaker,
is equivalent to a heaving thin ship of suitable shape. Using this result,
(181), and the equation shown in problem 11, prove that the amplitude of
the waves generated by a paddle-type wavemaker with horizontal velocity
U(y) cos ωt extending over the entire depth of deep water is given by
0
A=
2ω
∫ U ( y )e ky dy.
g −∞
Show from this result that a two-dimensional wavemaker, consisting of a
paddle that rotates with oscillatory velocity about a submerged pivot point,
will not generate waves at infinity if the depth of the pivot is suitably chosen in terms of the wavelength.
17. A vertical spar buoy of circular cylindrical form, draft T, and diameter d
is freely floating. Compute the hydrostatic restoring forces and moments.
Estimate the natural frequency in heave, assuming that the buoy is sufficiently slender that the added mass and damping coefficients can be
neglected by comparison to the mass of the buoy. Estimate the exciting
force from the Froude-Krylov approximation, the damping coefficient from
the Haskind relations, and compute the heave response. Compare your
answer with the results shown in figure 2.16.
18. A spherical floating platform of diameter 50 m is ballasted to float on
its equatorial plane and is unrestrained in a sea state corresponding to
a 40 knot Pierson-Moskowitz spectrum. On the same scale of frequency,
plot the heave response per unit wave amplitude from figure 6.24, the
338
Chapter 6
frequency spectrum of the incident waves, and the frequency spectrum of
the heave response of the buoy in these waves. Using graphical integration, find the average of the one-third highest heave amplitudes, the most
probable extreme value of the heave amplitude in 1000 wave encounters,
and the design extreme value of the heave amplitude in the same number
of waves.
References
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on the dynamics of marine vehicles and structures in waves. London: Institution of
Mechanical Engineering.
Cummins, W. E. 1962. The impulse response function and ship motions. Schiffstechnik 9:101–109.
Davis, A. M. J. 1976. On the short surface waves due to an oscillating, partially
immersed body. Journal of Fluid Mechanics 75:791–807.
Eggers, K., S. Sharma, and L. Ward. 1967. An assessment of some experimental
methods for determining the wavemaking characteristics of a ship form. Society of
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Gadd, G. 1976. A method of computing the flow and surface wave pattern around
full forms. Royal Institution of Naval Architects Transactions 118:207–216.
Graff, W., A. Kracht, and G. Weinblum. 1964. Some extensions of D. W. Taylor’s
standard series. Society of Naval Architects and Marine Engineers Transactions
72:374–403.
Haskind, M. D. 1953. Oscillation of a ship on a calm sea. English translation, Society
of Naval Architects and Marine Engineers. T & R Bulletin 1–12.
Havelock, T. H. 1963. Collected papers. ONR/ACR-103. Washington: U.S. Government
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Houmb, O. G., and T. Overik. 1976. Parameterization of wave spectra and long term
joint distribution of wave height and period. In Proceedings of the conference on the
behaviour of offshore structures (BOSS ’76). pp. 144–169. Trondheim: The Norwegian
Institute of Technology.
Kim, W. D. 1966. On a freely floating ship in waves. J. Ship Res. 10:182–191+200.
Kinoshita, M. 1943. Wave resistance of a sphere in a shallow sea. J. Society of Naval
Architects of Japan, 19–37.
Kinsman, B. 1965. Wind waves—their generation and propagation on the ocean surface.
Englewood Cliffs, N. J.: Prentice-Hall.
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Le Méhauté, B. 1976. An Introduction to hydrodynamics and water waves. New York,
Heidelberg, Berlin: Springer-Verlag.
Lewis, E. V. 1967. The motion of ships in waves. In Principles of naval architecture, ed.
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Lighthill, M. J. 1965. Group velocity. J. Institute of Mathematics and its Applications
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Longuet-Higgins, M. S. 1953. Mass transport in water waves. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
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hydrodynamics, eds. M. S. Plesset, T. Y. Wu, and S. W. Doroff, pp. 519–545. Washington: U.S. Government Printing Office.
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7 Hydrodynamics of Slender Bodies
Chapter
Hydrodynamics
of
Slender
7
Bodies
© Massachusetts Institute of TechnologyAll Rights Reserved
Many vessels of interest in marine hydrodynamics are slender, with one
length-dimension exceeding the others by an order of magnitude. For ships,
submarines, sailboats, and fish, this shape is generally a consequence of the
advantage of a streamlined body form, with the longitudinal length scale
substantially greater than the beam and depth. Slenderness also occurs for
various ocean platforms such as spar buoys and for bottom-mounted structures of similar form which are elongated vertically. For all of these vessels it
is logical to simplify the hydrodynamic analysis by suitable approximations
based on the slenderness of the body.
Slender-body theory originated in the field of aerodynamics, first as a
technique for predicting the stability characteristics of dirigibles. In ship
hydrodynamics, slender-body approximations have been utilized for a variety of problems, most of which have required substantial extensions or
revisions of the aerodynamic theory. The strip theory of ship motions in
waves is an important example, where the pioneering efforts of KorvinKroukovsky (1955) were motivated by parallel developments in aerodynamics, but the complexities of free-surface effects are such that the theory
is still in a state of refinement.
The applications of slender-body theory have been less intensive in
ocean engineering, although a strip theory can be used to analyze semisubmersible platforms supported by horizontal buoyancy hulls, as well as the
wave forces on slender vertical structures. In a very simple form, slenderbody theory has been applied successfully to spar buoys.
We shall begin our development of slender-body theory by discussing
two relatively simple problems—the longitudinal and lateral motion of a
slender body in an unbounded ideal fluid. These are problems that were
first treated in aerodynamics, and our study of the lateral motion will lead
342
Chapter 7
to a derivation of the low-aspect-ratio theory for planar lifting surfaces. The
results will be applied to the problem of ship maneuvering in the horizontal plane, where free-surface effects are generally small and the double-body
approximation can be used to avoid the complications of waves. Subsequently, wave effects on the free surface will be considered, first for a slender body with no forward velocity and then for a moving ship.
Slender-body approximations can be applied to analyze the effects of
shallow or restricted water on ships, as will be discussed in section 8, and
to predict the interactions between adjacent ships. The latter problem is
treated by Tuck and Newman (1974) and Yeung (1978). The applications
of slender-body theory to ship hydrodynamics are reviewed by Newman (1970) and Ogilvie (1974, 1977). A survey of ship hydrodynamics in
restricted waters is given by Tuck (1978).
7.1 Slender Body in an Unbounded Fluid
Let us consider the problem of a slender body, moving in an unbounded
ideal fluid, with the fundamental assumption that the transverse dimensions of the body are small compared to its length. To formalize this
assumption, we define the slenderness parameter
d / l = ε,
(1)
where d is the maximum lateral dimension of the body, and l is its length.
The nondimensional parameter ε is assumed small, and on this basis an
approximate solution is sought for the hydrodynamic quantities of interest.
The body motions are assumed to consist of a constant forward velocity
U, parallel to the x0-axis of a space-fixed coordinate system (x0, y0, z0), and a
small lateral motion in the z0-direction. (Vertical motions in the y0-direction
can be treated in an identical manner.) A moving coordinate system (x, y, z)
is defined by the transformations
x = x0 − Ut ,
(2)
y = y0 ,
(3)
z = z0 .
(4)
The lateral motion of the body is described by its displacement ζ(x, t)
from the x0-axis, as shown in figure 7.1. For rigid-body motions ζ will be a
Hydrodynamics of Slender Bodies
343
constant or linear function of x. Undulatory motions of the body may also
be considered where the dependence of ζ on x will be more general. We
assume only that ζ is small relative to the body length and that this displacement is a slowly varying function of x. The latter restriction excludes
flexural modes with a length-scale of order εl.
The body surface SB is assumed to be elongated in the longitudinal
direction and may include a tail fin as shown in figure 7.1. The intersection of SB with a plane x = constant defines the cross-section profile ∑B(x),
and the circumscribed area S(x) is the sectional area of the body. We shall
assume that the body ends are such that both S(x) and S′(x) ≡ dS/dx vanish at
the ends.
The body surface can be described by the equation F(x, y, z − ζ) = 0. For
example, if the body is symmetrical about z = ζ, then F = z – ζ ± b(x, y) where
b is the local half-beam. Using the substantial derivative (3.19), the boundary condition on SB takes the form
D
F( x0 − Ut , y 0 , z0 − ζ ) = 0,
Dt
(5)
or
∂ζ ∂ζ  ∂F
 ∂φ
 ∂F ∂φ ∂F  ∂φ ∂ζ ∂φ
= 0.
−U
+
+
−
+U
− 
 ∂x0
 ∂x ∂y 0 ∂y  ∂z0 ∂x ∂x0
∂x ∂t  ∂z
(6)
Here ϕ(x0, y0, z0, t) is the velocity potential due to the presence of the
body, which is defined initially with respect to the fixed coordinates
(x0, y0, z0).
Figure 7.1
Definition sketch for slender body and coordinate systems.
344
Chapter 7
Since ∇F is a vector normal to SB, (6) can be rewritten in the form
∂φ
∂ ζ ∂ φ ∂ζ 
 ∂ζ
= Unx + 
−U
+
 nz , on SB ,
 ∂t
∂n
∂x ∂x0 ∂x 
(7)
where n is the unit normal vector into the body. The boundary-value problem is completed by requiring that the potential vanish at large distances
from the body,
φ→0
as ( x2 + y 2 + z 2 )1/2 → ∞,
(8)
and by invoking Laplace’s equation,
∂ 2φ ∂ 2φ ∂ 2φ
+
+
= 0,
∂x20 ∂y20 ∂z20
(9)
throughout the fluid domain.
Before making the slenderness approximation, we note that the first
term on the right side of (7) is associated with the boundary condition
for the longitudinal-flow problem, and the remaining contribution to (7)
is associated with the lateral motion. These two problems can be treated
separately, since the lateral motion is assumed small, with a linear decomposition identical to the lifting and thickness problems treated in chapter
5. In particular, the boundary condition for the longitudinal-flow problem
can be imposed on the “stretched-straight” position of the body with ζ = 0.
The boundary-value problem (7–9) can be simplified if the body is slender. Geometrically, it follows from (1) that the unit normal vector n will
coincide with the plane x = constant as ε → 0, or
nx = O( ε ),
ny = O(1),
nz = O(1).
(10)
On this basis, the normal derivative in (7) can be approximated by the twodimensional normal derivative in the y-z plane.
There is also a hydrodynamic consequence of slenderness. To see this, we
adopt an inner reference frame, fixed with respect to the scale of the cross
section ∑B. From this viewpoint, d = O(l) whereas l = O(l/ε). As ε → 0, the
transverse scale remains fixed while the body length tends to infinity. The
fluid velocity field then appears nearly constant in the longitudinal direction, with ∂/∂x0 = O(ε). In this inner region it follows that
∂φ
 ∂φ ∂φ 

,
,
 ∂y 0 ∂z0 
∂x0
(11)
Hydrodynamics of Slender Bodies
345
with a similar result for the second derivatives. Thus Laplace’s equation
reduces to the two-dimensional form
∂ 2φ ∂ 2φ
+
= 0,
∂y20 ∂z20
(12)
and ϕ can be replaced by a two-dimensional potential, say
φ = Φ ( y 0 , z0 ; x0 ).
(13)
Here the dependence on x0 is included to emphasize that this potential will
vary slowly along the body length, as a result of the change in the body
geometry and lateral motion.
Since ζ and ε are both small, the body potential must be small in some
sense, and if the forward velocity U = O(l), then
∂φ
U.
∂x0
(14)
With these approximations the boundary condition (7) can be replaced by
∂Φ
∂ζ
∂ζ 
= Unx + 
−U
nz , on Σ B .
 ∂t
∂N
∂x 
(15)
Here N denotes the two-dimensional unit vector normal to ∑B in the y-z
plane. Note that the longitudinal-flow term Unx has been retained, in spite
of (10), since ε and ζ are independent small parameters.
From the boundary condition (15) and the linearity of the boundaryvalue problem, it is logical to express the solution as the sum of two
potentials, one due to the longitudinal motion and the other to the lateral
motion. With a notation suggested by (4.101–102), we shall write
Φ = U Φ 1 + WΦ 3 ,
(16)
where, on ∑B
∂Φ 1
= nx ,
∂N
(17)
∂Φ 3
= nz ,
∂N
(18)
and
346
Chapter 7
W ( x, t ) =
∂ζ
∂ζ
.
−U
∂t
∂x
(19)
Physically, W(x, t) is the lateral velocity of the body section, as observed
in a fixed frame of reference. A similar form may be recalled for the boundary condition (5.193) on an unsteady lifting surface. The potentials Φ1 and
Φ3 satisfy the two-dimensional Laplace equation and the boundary conditions (17–18), and they correspond to the solutions of two-dimensional
flow problems at each section along the body length.
The lateral-flow problem for the potential Φ3 can be identified precisely with the lateral translation of a two-dimensional body, with profile
∑B, moving in the positive z-direction with velocity W. In the special but
important case of a body of revolution, with local radius R(x), the solution
follows from table 4.2 in the form
Φ 3 = −( R2 / r )cos θ ,
(20)
where r = (y2 + z2)1/2 and θ = tan−1(y/z). This solution holds at each section
along the body length, where R(x) is equal to the local radius. For other
body profiles the solution can be obtained by conformal mapping or other
techniques. The lateral force acting on the body at each section is related
to the two-dimensional added-mass coefficient in a manner which will be
derived in section 7.3.
The longitudinal flow and the two-dimensional potential Φ1 are more
complicated. The boundary condition (17) does not correspond to a twodimensional rigid-body motion but to a dilation of the body profile. Integrating (17) around the body section to obtain the local flux Q, we get
∂Φ 1
dl = − ∫ nx dl = − S′,
∂N
ΣB
ΣB
Q = −
∫
(21)
where S′ denotes the derivative of the sectional area S(x). Physically, (21)
may be anticipated from the fact that longitudinal motion of the body
displaces the surrounding fluid, with outward flux equal to the change in
sectional area of the body.
For large radial distances from the body axis, the two-dimensional
potentials are of the general form (4.70), and from flux considerations Φ1
must include a source term with strength −S′. Thus,
Φ1 −
S′
log( r / l ),
2π
for r d ,
(22)
Hydrodynamics of Slender Bodies
347
where the length l has been inserted for nondimensional purposes. On the
forebody Φ1 will be source-like, and on the afterbody sink-like, with corresponding radial outflow and inflow.
It is inevitable that as r → ∞, the two-dimensional solution (22) will
diverge and cannot be reconciled with (8). This incompatibility is associated with the assumption of a two-dimensional flow. In particular, the argument leading to (11) was made with the distance from the body fixed as
its length tends to infinity. It is not appropriate to apply (8) in this inner
region, nor to the inner solution Φ1, since (8) holds only at large distances
compared to all the body dimensions, including the length.
Since the condition (8) cannot be applied to the inner solution, the
potential Φ1 contains an arbitrary constant not determined either from
Laplace’s equation or from the boundary condition on the body. Thus,
at this stage, the inner solution is not unique, nor is it well-behaved at
infinity.
These deficiencies result from the assumption of two-dimensional
flow, and thus a three-dimensional solution of the Laplace equation (9) is
required to bridge the gap between the inner solution Φ1 and the condition at infinity. The appropriate outer solution is a distribution of threedimensional sources, along the body axis, which satisfy (8) at infinity
and can be matched with (22) near the body. The latter condition serves
to determine the strength of the three-dimensional source distribution;
from this outer solution, the arbitrary constant of the inner solution is ultimately determined. The approach outlined here is essentially the method
of matched asymptotic expansions referred to earlier in connection with
the boundary-layer approximation and lifting-line theory. The common
feature of these problems and slender-body theory is the disparity in length
scales that characterize the flow. A more extensive discussion of such problems is given by Van Dyke (1975).
The method of matched asymptotic expansions will be employed in section 7.2 to solve the longitudinal-flow problem. Subsequently, in section
7.3, we shall return to the lateral-flow problem, which is in some respects
simpler and more useful, to analyze the lateral force acting along the body
length.
Before treating the longitudinal-flow problem in detail, it should be
noted that the need for a three-dimensional outer solution does not arise
for the lateral flow. Essentially this distinction occurs because no net flux
348
Chapter 7
is associated with the boundary condition (18) for lateral motion, and thus
there is no source term in the local solution of the lateral flow problem.
The potential Φ3 behaves for large r as a two-dimensional dipole, as in (20)
for a body of revolution or (4.129) for arbitrary body profiles. This dipole
potential vanishes at infinity, and thus is consistent with (8). Strictly, this
behavior is not identical to the more appropriate three-dimensional dipole
(4.123), but the distinction does not affect the leading-order slender-body
solution for the lateral-flow problem.
7.2 Longitudinal Motion
The longitudinal problem, associated with the steady forward motion
of the body, can be solved most readily in the moving coordinate system
(x, y, z). Here the body is in its stretched-straight position and is rigid. In
accordance with (4.101), the potential due to the body disturbance will be
of the form
Φ = U φ1( x, y , z ).
(23)
We shall seek a solution of this problem using the method of matched
asymptotic expansions, where the inner and outer solutions are treated as
complementary approximations, each of which is valid in its own domain.
Thus the inner solution Φ1 is governed by the two-dimensional Laplace
equation (12) and the body boundary condition (17), but not by the condition (8) at infinity. Conversely, the outer solution φ1 is governed by the
three-dimensional Laplace equation and by the condition (8) at infinity,
but need not satisfy the body boundary condition. These two solutions will
be consistent provided they can be matched in an overlap region εl ≪ r ≪ l
far from the body in the inner region but very close to the body in the outer
region. Thus the matching requirement
ϕ1 Φ1 ,
εl r l ,
(24)
effectively replaces the missing boundary condition at infinity in the inner
problem and at the body in the outer problem.
This method is useful because the two separate problems are simpler
to solve, compared to the exact solution valid everywhere throughout
the fluid. The inner solution is simplified because it is two dimensional,
Hydrodynamics of Slender Bodies
349
whereas the outer solution is simplified by replacing the body boundary
condition with (24).
Combining (22) and (24) we get the inner condition for φ1 in the form
ϕ1 −
1
S ′ log( r / l ),
2π
r l.
(25)
The appropriate outer solution is a source distribution, along the body
length, with the same strength as (25). Using the three-dimensional source
potential (4.31), and distributing these along the body axis gives
ϕ 1 ( x, r ) =
1
S ′(ξ )[( x − ξ )2 + r 2 ]−1/2 dξ.
4π ∫l
(26)
To confirm that (26) can be matched to the inner solution, it is necessary
to develop the inner expansion of (26) for r/l ≪ 1. For this purpose we shall
use the relation
[( x − ξ )2 + r 2 ]−1/2 =
1/2
 ξ − x  ξ − x  2
 
d
+ 1  ,
log 
+ 

dξ
 r 
 
 r
(27)
which can be verified directly by differentiation. Substituting (27) in (26)
and integrating by parts gives
ϕ1 = −
1/2
 ξ − x  ξ − x  2
 
1
+ 1  dξ.
S ′′(ξ )log 
+ 

∫
4π l
 r 
 
 r
(28)
Here the contributions from the endpoints are deleted on the assumption
that S′ = 0 at the ends.
To approximate (28) for small r, we note that the limiting value of the
argument of the logarithm is given by
1/2
2
  2(ξ − x)
 ξ − x  ξ − x 
+ 1  + 
, (ξ − x ) > 0 ,


r
 r 
 
 r
(29)
as r → 0. Similarly, from a Taylor series expansion of the square-root function, or by noting from (27) that the logarithm must be an odd function of
(x − ξ), it follows that
1/2
 ξ − x  ξ − x  2
 
r
+ 1  + 
, ( ξ − x ) < 0.


 r 
  2 ξ − x
 r
(30)
350
Chapter 7
Substituting (29–30) in (28), the inner expansion of the outer solution for
small r is obtained in the form
ϕ1 1
4π
x
 2( x − ξ )  dξ − 1
4π
r

∫ S′′(ξ )log 
xT
xN
 2( x − ξ )  dξ.
r

∫ S′′(ξ )log 
x
(31)
Here xN and xT denote the x-coordinates of the nose and tail. Separating the
terms involving log r and integrating these, it follows that
ϕ1 −
1
S ′ log( r / l ) + f ( x),
2π
r l.
(32)
where
f ( x) =
1
4π
x
∫
S ′′(ξ )log[ 2( x − ξ ) / l ]dξ −
xT
1
4π
xN
∫ S′′(ξ )log[ 2(ξ − x) / l ]dξ.
(33)
x
Equation (32) is the inner expansion of the outer solution, which contains the desired two-dimensional source behavior, consistent with (25).
However, a “constant” term (33) is left over; it is independent of the (y, z)
coordinates but depends on the local value of x. Since the inner solution
contains an arbitrary constant, the matching requirement dictates the
value of this constant to be equal to (33), and the inner solution is uniquely
determined by the requirement that
Φ1 −
1
S ′ log( r / l ) + f ( x),
2π
r ε l.
(34)
Note that the body need not be axisymmetric, and the complexities of
its geometry will be reflected in the inner solution Φ1. The outer limit of
this solution (34), and thus the outer solution φ1, are axisymmetric, but
that is a consequence of the source-like flow far from the body, where its
detailed shape is unimportant. For a body of revolution, (34) is the inner
solution for all values of r. For other body profiles, the inner solution must
be found by conformal mapping or similar techniques.
This completes the solution of the longitudinal-flow problem for a slender body in an infinite fluid, and a brief review of the derivation may be
useful. We began in section 1 with an assumed inner solution, valid near
the body surface and satisfying the body boundary condition and the twodimensional Laplace equation. However, this solution does not satisfy the
Hydrodynamics of Slender Bodies
351
condition at infinity in the outer field, and it contains an arbitrary additive constant. To provide a corresponding outer solution with the proper
behavior at infinity, a three-dimensional source distribution is assumed,
using flux conditions to guess the appropriate source strength. By examining the inner approximation of this outer solution, for small radius r, the
outer solution can be matched to the inner solution provided the arbitrary
constant of the latter is chosen as defined in (33). Thus, in the process of
overcoming the deficiencies at infinity, the arbitrary constant of the inner
solution is prescribed.
The “constant” (33) is neither disposable nor unimportant, since it affects
the value of the pressure distribution along the body surface. To illustrate
this, we note that the added-mass coefficient m11 may be computed from
the inner solution, using (4.114). For a body of revolution it follows that
1
∂Φ 1
m11 = ρ ∫∫  −
S ′ log( r / l ) + f ( x )
dS
π
∂n
2


SB
1
S ′ log( R / l ) + f ( x) S ′( x) dx.
= ρ ∫∫  −
 2π

(35)
l
Since the body radius R is of order εl, the added-mass coefficient for
longitudinal motion is proportional to l3ε4 log ε. By comparison, the body
volume is of order ε2l3. Thus, the added mass of a slender body in the longitudinal direction is negligible compared with the body mass. This conclusion is not restricted to bodies of revolution, as shown in problem 1.
7.3 The Lateral Force
To find the distribution of the lateral force along the body length, (4.90)
may be applied to the thin slice of fluid of differential thickness dx0, as
shown in figure 7.2. This volume of fluid is bounded by the differential
length of the body surface SB, of profile ∑B (x) and width dx0, as well as by
a fixed control surface SC. The surface SC is composed of two lateral planes
S0 separated by a distance dx0 exterior to the body, and by a closure surface
S∞, at large radial distance from the body. Applying (4.90) to this control
surface, as well as to the differential portion of the body surface, the differential force is given by
352
Chapter 7
Figure 7.2
Control volume for the evaluation of the differential lateral force Fz′.
Fz′ = − ρ
d
dt
∫
Σ
Φ nz dl − ρ
B ( x)
∂
∂x0
∂φ ∂φ
dy 0 dz0
0 ∂z 0
∫∫ ∂x
S0
1
 ∂ϕ ∂φ

− nz ∇φ ⋅ ∇φ  dl.
= −ρ ∫ 


n
2
∂
z
∂
0
Σ∞
(36)
Here the differential operator d/dt is to be interpreted as the time-derivative
in a fixed reference frame, with x0 = constant, ∑∞, is the profile defined by
the limit of S∞, as dx0 → 0, and S0 is the plane bounded by ∑B (x) and ∑∞.
From the slenderness assumption (11) there is no contribution from the
integral over S0, and since the potential vanishes at infinity there is no
contribution from the last integral in (36). Thus the differential lateral force
acting along the body length is given by the expression
Fz′ = − ρ
d
dt
∫
φnz dl.
(37)
ΣB ( x)
On the body, the potential can be replaced by (16), and if the profile
is symmetrical about z = 0, there is no contribution to the integral in (37)
from Φ1. Thus it follows that
Hydrodynamics of Slender Bodies
Fz′ = − ρ

d 
d
Φ 3nz dl  = − (W ( x, t )m33 ( x)) .
W 
∫
dt  Σ B ( x )
dt

353
(38)
Here the two-dimensional added-mass coefficient m33(x) has been introduced, using the relation (4.114) and the boundary condition (18). In view
of the coordinate transformation (2), the time-derivative in (36–38) can
be replaced by partial derivatives in a moving reference frame, and (38) is
equivalent to
∂
∂
Fz′ = −  − U  [W ( x, t )m33 ( x)].
 ∂t
∂x 
(39)
Equation (39) has been derived by Lighthill (1960), in analyzing the
swimming motion of slender fish. Various special cases were known earlier in the field of aerodynamics. One feature of (39) and of the derivation we have carried out thus far is that the time-dependence is relatively
unimportant, and the distinction between steady and unsteady problems is
trivial.
The total force acting on the body can be obtained in a strip-theory manner by integrating (39). The lateral added-mass coefficient, due to acceleration of a rigid slender body in the z-direction, is given by the integral
s
m33
= ∫ m33 ( x) dx.
(40)
l
Similarly,
s
m35
= ∫ m33 ( x) xdx,
(41)
l
and
s
m55
= ∫ m33 ( x)x2 dx,
(42)
l
s
s
where m55
is the added moment of inertia for yaw acceleration, and m35
denotes the cross-coupling added mass between sway and yaw. The latter
vanishes if the body is symmetrical about x = 0.
The superscript s in (42) is inserted to distinguish these strip-theory
results. In this approximation, the local force at any section of the body is
not affected by the shape of the body elsewhere; that is, there are no hydrodynamic interactions between adjacent sections of the body. The resulting
354
Chapter 7
simplification is significant. For example, the results for a slender spheroid
can be deduced in terms of the added-mass coefficient for a circle without the complexity of solving the complicated three-dimensional problems
represented by figure 4.8. Of course, a price must be paid for this simplification, and it is apparent from figure 4.8 that the sway added-mass coefficient
will be overpredicted in the strip theory by a factor of 10 percent when the
diameter-length ratio is 0.2, and by 100 percent in the extreme case of a
sphere. A larger error results for the added moment of inertia, as emphasized by the fact that this coefficient vanishes for a sphere but is nonzero in
the strip theory! (See problem 2.) It would be naive to suggest that a sphere
is slender, and in general a diameter-length ratio of 0.1 to 0.2 is a reasonable
upper limit for the slender-body approximation.
The total force acting on a slender body in steady motion can be derived
by integrating the differential force (39),
Fz = U ∫
l
∂
[W ( x)m33 ( x)] dx = U [W ( x)m33 ( x)]xxTN .
∂x
(43)
For a slender body, the nose is a point of zero transverse dimensions, and
thus m33(xN) = 0. The same will be true at the tail for a pointed body, without
a tail fin; in accordance with D’Alembert’s paradox, the lateral force acting
on such a body in steady motion is zero.
For a body with a tail fin, such as that shown in figure 7.1, the situation
is fundamentally different. Using the flat-plate result from table 4.3, the
added-mass coefficient of the tail is given by
m33 ( xT ) =
π 2
ρs ,
4
(44)
where s is the tail span. Using (19) to evaluate W, and denoting the slope
∂ζ/∂x at the tail by the local angle of attack αT, it follows that there will be
a transverse “lift” force
Fz =
π
ρU 2 s2α T .
4
(45)
The yaw moment about the vertical axis of the body can be obtained
from (39) in a manner similar to that used to obtain the force. For steady
motion, we obtain the following expression in place of (43):
Hydrodynamics of Slender Bodies
M y = −U ∫ x
l
355
∂
[W ( x )m33 ( x)] dx
∂x
= U ∫ W ( x )m33 ( x ) dx + UxT W ( xT )m33 ( xT ).
l
Using (19) for W, and (45) in the last term,
M y = −U 2 ∫
l
∂ζ
m33 ( x) dx − xT Fz .
∂x
(47)
The last term is associated with the moment due to the lift force on the tail
fin and vanishes for a body with a pointed tail. The remaining integral in
(47) gives a contribution for a body moving at an angle of attack, regardless
of whether the tail is pointed. This Munk moment acts on a nonlifting body
in steady translation, as noted in section 4.13.
The orders of magnitude of the velocity potential Φ3 and lateral force
F3, in terms of the slenderness parameter ε, can be deduced from the above
expressions. In terms of the inner solution of the problem, where the length
scales are referred to the transverse dimensions of the body and where the
body length is very large, the two-dimensional quantities Φ3 and F3' are
quantities of order one.
On the other hand, if the body length l = O(1) and the transverse dimensions are regarded as O(ε), then the potential Φ3 is O(ε) and the lateral force
is O(ε2). The latter results can be deduced from dimensional arguments,
based on the fact that these are quantities of order one in the inner frame
of reference. From either standpoint, the lateral force is of the same order as
the body mass, and both will be of comparable importance in the equations
of motion for an unrestrained body subject to a lateral disturbance. This
situation contrasts with that of the longitudinal force.
This analysis of the lateral force on a slender body is based on the assumption that the flow is irrotational at each section along the body length.
Thus, vortex sheets and separation are excluded in the fluid alongside the
body. However, the flow downstream of the body does not affect this analysis, and a trailing vortex sheet may originate from the abrupt trailing edge
at the tail. From the viewpoint of lifting-surface theory, this vortex sheet is
associated directly with the lift force (45).
These results can be applied to planar lifting surfaces of small aspect
ratio, where the body profile ∑B is a flat plate of local span s(x), provided no
vorticity is shed upstream of the tail. This restriction requires the trailing
356
Chapter 7
edge to be abrupt and situated at the tail. In addition, the angle of attack
must be sufficiently small to avoid leading-edge separation. Under these
conditions, the results above reduce to the low aspect ratio lifting-surface
theory. In the notation of chapter 5, the lift coefficient follows from (45)
in the form
CL = 21 π Aα ,
(48)
where A = s2/S is the aspect ratio based on the maximum span and planform
area S. As shown in figure 5.22, (48) gives precisely half the limiting value,
for A ≪ 1, of the lifting-line theory (5.148). This difference is not surprising, since the lifting-line theory assumes that A ≫ 1.
The longitudinal distribution of the lift force on an uncambered lifting
surface of small aspect ratio is proportional to the rate of change of added
mass. For a “delta wing” with triangular planform, the lift force will be distributed uniformly along the length. If the span increases more rapidly, the
center of pressure will move forward. In the limiting case of a rectangular
planform, with constant span along the length, the lift force will be concentrated at the leading edge, with a differential lift given by
Fz′ =
π
ρU 2α s2δ ( x − xN ),
4
(49)
where δ(x − xN) denotes a delta function. This limit is physically unreasonable, but experiments show that the center of pressure for a rectangular
lifting surface of small aspect ratio is very close to the leading edge, particularly in the regime of small angles of attack where leading-edge separation
does not occur.
For a planar lifting surface of small aspect ratio with the maximum span
at the tail, the trailing vortex sheet downstream of the body must be independent of the coordinate x, with a constant downwash velocity equal to
UαT, Thus, it makes no difference whether the body terminates at the position of maximum span or continues downstream with decreasing span,
provided the local angle of attack is constant. For this reason, the low aspect
ratio theory holds for uncambered lifting surfaces of more general planform,
if the maximum span is used in (48). In this particular case, there is no lift
force on the portion of the body downstream of the maximum span.
Hydrodynamics of Slender Bodies
357
The extension of slender-body theory to account for the interaction of
the afterbody with vortex sheets shed upstream has been carried out by
Newman and Wu (1973) in the general case where the local lateral velocity of the body differs from the downwash of the trailing vortices. Vortex
sheets that occur due to separation are analyzed by Fink and Soh (1974). In
both cases, the vorticity along the body is convected with the free-stream
velocity U, and “memory” effects will occur in the unsteady case as in the
two-dimensional lifting-surface theory.
7.4 Ship Maneuvering: The Hydrodynamic Forces
The hydrodynamic forces of significance during a ship maneuver are primarily the sway force and yaw moment. Anticipating large excursions in
the horizontal plane, as in the turning trajectory of a ship, it is customary
to employ a coordinate system fixed with respect to the ship. In this frame
of reference, the forward velocity is of order one, whereas the sway and yaw
velocities are assumed small by comparison. The remaining three modes of
rigid-body motion are ignored, on the presumption that changes in the roll
angle, heave, and pitch of the ship will be unimportant.
We shall employ the same coordinates and notation here as in chapter
4. The body is assumed rigid, and the (x, y, z) axes are fixed with respect to
the ship such that x is positive forward, y upward, and z to starboard. The
three nonzero velocity components of the ship are U1 (forward), U3 (sway),
and Ω2 (yaw). These are indicated in figure 7.3, together with the sway force
F3 and yaw moment M2, as well as the rudder angle δR defined to be positive
for a turn to port.
Figure 7.3
Horizontal motions of a ship as seen from above.
358
Chapter 7
This convention differs from the standard throughout the field of ship
maneuvering, where the y-axis is positive to starboard and the z-axis is
directed vertically downward. In the standard notation, as described by
Mandel (1967), the sway velocity v and force Y are identical to U3 and F3,
but the yaw velocity r and moment N differ in sign from Ω2 and M2.
Ship maneuvers are relatively slow, compared to vertical motions in
waves, due to the physical limitations of the control systems involved. Thus
it is reasonable to assume that the sway velocity U3 and yaw velocity Ω2 are
slowly changing functions of time and small by comparison to the scale
of the forward speed. Neglecting wave effects associated with the steady
forward velocity, the lateral flow associated with the sway and yaw motions
can be analyzed on the basis of the low-frequency limit of the free-surface
condition; that is, wave effects are neglected and the free surface is replaced
by a rigid horizontal plane. The appropriate image of the ship hull can be
inferred from figure 6.21, and for lateral motions it is appropriate to treat
the wetted portion of the hull as the lower half of a rigid double body, as
shown in figure 7.4.
In addition, it will be assumed that the fluid motion is ideal, and thus
viscous effects can be ignored. Boundary layer phenomena and the related
frictional drag are of no significance here, but if the angle of attack between
the ship hull and the water becomes large, viscous separation will occur in
the cross-flow past the hull. This is a nonlinear phenomenon, beyond the
scope of our analysis, which must be regarded as a possible limitation of
the theory.
With these preliminaries, we proceed to analyze the ship hull and its
image as a rigid double body, of slender form, which can develop a lateral
lift force in accordance with the results of section 3. Since U3 and Ω2 are the
sway and yaw velocities defined with respect to the body fixed reference
frame, the total lateral velocity is
Figure 7.4
Elevation of the ship hull and its double-body image above the free surface.
Hydrodynamics of Slender Bodies
W ( x, t ) = U 3 (t ) − xΩ2 (t ).
359
(50)
It follows from (39) that the local force acting along the hull is given by
∂
∂
Fz′ = −  − U 1  [(U 3 − xΩ2 )m33 ( x )]
 ∂t
∂x 
∂
∂
= − (U 3 − xΩ 2 ) m33 ( x ) + U 1U 3 ( m33 ) − U 1Ω2 ( xm33 ).
∂x
∂x
(51)
Integrating along the hull length gives the total sway force,
S
S
F3 = −U 3m33
− Ω 2 m35
− U 1U 3m33 ( xT ) + U 1Ω2 xT m33 ( xT ).
(52)
Here the superscript s denotes the strip-theory added-mass coefficients
(40–42), and xT is the value of x at the effective trailing edge, nominally
the stern.
Similarly, multiplying (51) by the moment arm −x and integrating over
the length gives the yaw moment
S
S
S
M 2 = −U 3m35
− Ω 2 m55
+ U 1U 3[ m33
+ xT m33 ( xT )]
S
+ U 1Ω2[ m35
− ( xT )2 m33 ( xT )].
(53)
Equations (52–53) give the sway force and yaw moment acting on the
hull in accordance with the slender-body results of section 7.3. In table
7.1 these expressions are compared with the corresponding equations
which follow from (4.115–4.116), where no assumption of slenderness is
made but where the body must be pointed at the stern and without lifting
effects.
From a comparison of the results shown in table 7.1, the slender-body
theory neglects the longitudinal added-mass coefficient m11, which is consistent with our conclusion in section 7.2 that this coefficient is of higher
order in the slenderness parameter ε. On the other hand, the slender-body
theory includes the lifting force and moment associated with the action
of the double body as a lifting surface of low aspect ratio. The physical
relevance of these complementary descriptions depends on the nature of
the ship’s stern. Generally, the stern profile is as shown in figure 7.4, and it
is not obvious whether this should be regarded as pointed or with a sharp
trailing edge. However, the actions of the rudder and deadwood are significant in practice, and it is likely that additional vorticity is shed from the
keel along the afterbody in a manner which coincides more nearly with the
slender-body results.
360
Chapter 7
Table 7.1
Hydrodynamic Coefficients in Equations (52–53), Based on the Slender-Body Theory
with Lifting Effects Included
Present
Notation
Standard
Notation
Slender-Body
Theory
Added-Mass
Theory
∂F3/∂U3
Yυ
−U1mT
0
∂F3/∂Ω2
∂F3 / ∂U 3
−Yr
U1xTmT
U1m11
Yυ
−m
−m33
∂F3 / ∂Ω 2
−Yr
S
−m35
∂M2/∂U3
−Nυ
U 1[ m + xT mT ]
U1(m33 − m11)
∂M2/∂Ω2
∂M2 / ∂U 3
Nr
S
U 1[ m35
+ ( xT )2 mT ]
U1m35
−Nυ
−m
−m35
N r
−m
−m55
∂M2 / ∂Ω 2
S
33
−m35
S
33
S
35
S
55
Note: The second column is the standard notation described by Mandel (1967), and
the last column shows the coefficients that follow from equations 4.115–4.116). The
superscript s denotes the strip-theory approximations, and mT denotes the addedmass coefficient at the stern.
To determine the validity of the slender-body theory, rough estimates
of the force coefficients shown in table 7.1 can be compared with experiments. To facilitate these estimates we shall assume that the draft T is
constant along the length, and that the two-dimensional added-mass coefficient can be approximated by its value for a flat plate or ellipse. Utilizing
the results from table 4.3, with a factor of 1/2 to include only the lower half
of the double body, it follows that
m33 ( x) = 21 πρT 2 .
(54)
Taking the origin at the midship section, xT = −l/2 and the results shown in
table 7.2 follow from table 7.1 and equation (54).
The most significant discrepancy in table 7.2 occurs for the sway force
due to a constant sway velocity, where the theory underpredicts the experiments by about 50 percent. Here the importance of viscous cross-flow drag
is recalled. In this connection, Norrbin (1970) notes that the experimental
value of this force coefficient is sensitive to nonlinear effects and the interpretation of the experimental results. Nor does the theory account for the
cross-coupling yaw moment due to sway acceleration, due to our simplifying assumption concerning the planform and added-mass distribution.
Hydrodynamics of Slender Bodies
361
Table 7.2
Theoretical and Experimental Values of the Coefficients Listed in Table 7.1, for a
Mariner Ship Hull with Length-Draft Ratio l/T = 21
Slender-Body Theory
for Rectangular
Planform
Theory for L/T
of Mariner Hull
(× 103)
Experiments for
Mariner Hull
(× 103)
−13.3 ± 3.6
∂F3/∂U3
−
π
ρU 1T 2
2
−7.1
∂F3/∂Ω2
−
π
ρU 1T 2l
4
−3.6
−2.62 ± 0.72
∂F3 / ∂U 3
−
π
ρT 2l
2
−7.1
−6.55 ± 0.93
∂F3 / ∂Ω 2
0
0
0.25 ± 0.08
∂M2/∂U3
π
ρU 1T 2l
4
3.6
3.69 ± 0.78
∂M2/∂Ω2
−
−1.8
−2.40 ± 0.50
∂M2 / ∂U 3
0
0
0.22 ± 0.08
∂M2 / ∂Ω 2
−
−0.6
−0.36 ± 0.12
π
ρU 1T 2l 2
8
π
ρT 2l 3
24
Note: The coefficients in the last two columns are nondimensionalized in terms of
the quantities 21 ρ , U 1 , and l. The theoretical values are based on the slender-body
results shown in table 7.1, but with the simplifying assumptions that the draft T is
constant along the length, and the added-mass coefficient can be approximated by
(54). References to the experimental data are given by Mandel (1967), and Motora
(1972).
Finally, the theory overpredicts the added moment of inertia in yaw by a
substantial amount; again this may be attributed to our assumption of a
rectangular planform with excessive added mass at the bow and stern.
Despite these differences, the slender-body theory gives a satisfactory
qualitative prediction of the hydrodynamic force and moment. On this
basis it will be used to study the maneuvers of a ship in the horizontal
plane.
The rudder force can be predicted from the slender-body theory, on the
premise that the rudder-hull combination is a single cambered lifting surface. With reference to figure 7.1, the action of the rudder can be regarded
in the slender-body theory as a deflection of the hull equal to
ζ ( x, t ) = δ R (t )( x − xR ),
(55)
362
Chapter 7
for xT < x < xR. Here xR denotes the longitudinal coordinate of the rudder
axis. The contribution from the rudder to the lateral velocity (50) is
W ( x, t ) = δ R ( x − xR ) − U 1δ R .
(56)
Substituting (56) in (39) and integrating the resulting force from upstream
of the rudder post to its trailing edge gives the rudder force
Fz = mT δRlR2 / 2 + 2U 1lRδ R + U 12δ R  .
(57)
Here mT has been substituted on the premise that the two-dimensional
added mass of the rudder is independent of position along its chord, which
is denoted by lR. Neglecting the small change in moment arm from the origin to different positions on the rudder, we may take the yaw moment as
the product of (57) and −xT.
The terms in (57) proportional to the angular velocity and acceleration
of the rudder are unsteady effects that can be ignored in practice. These are
significant only over time scales of order lR/U1, the time required for the
ship to travel one chord length of the rudder, which is negligible compared
to the response time of the ship.
Retaining only the steady term in (57) gives the rudder force and moment
in the form
F3 R = 21 πρT 2U 12δ R ,
(58)
M 2 R = − 21 πρT 2U 12δ R xT .
(59)
Here (54) has been used for the added-mass coefficient.
The results (58–59), for the rudder force and moment, overpredict the
experimental values reported by Motora (1972) by a factor of two. This discrepancy may occur because the span of the actual rudder is significantly
less than the draft T.
As an alternative to the use of (58–59), the rudder can be regarded as a
separate appendage, with hull interactions neglected. The lifting surface
theory or experimental data can be used to estimate the rudder force and
moment, in a manner outlined by Mandel (1967). In practice there are
significant interactions with the hull and with the slipstream from the propeller if this is upstream of the rudder. It is difficult to predict these interactions from model tests with Froude scaling, since the Reynolds number of
the rudder is reduced significantly from its full-scale value.
Hydrodynamics of Slender Bodies
363
7.5 Ship Maneuvering: The Equations of Motion
In the absence of external forces, the sway and yaw motions of a ship can be
determined by equating the hydrodynamic force and moment derived in
the preceding section to the body-mass force and moment. The body-mass
contributions can be obtained by replacing the added-mass coefficients
in table 7.1 by the body-mass coefficients Mij defined in equation (4.141).
Using the slender-body results of table 7.1 for the hydrodynamic forces on
the hull, and using (58–59) to estimate the rudder force and moment, the
equations of motion follow in the form
1
2
πρT 2U 12δ R = U 1mTU 3 − U 1( xT mT + m )Ω2
S
S
+ ( m33
+ m )U 3 + ( m35
+ M35 )Ω 2 ,
(60)
S
− 21 πρT 2U 12δ R xT = −U 1( m33
+ xT mT )U 3
S
− U 1[ m35
+ M 35 − ( xT )2 mT ]Ω2 ,
(61)
S
+ M 55 )Ω 2 .
+ ( m3S5 + M 35 )U 3 + ( m55
Equations (60–61) are a pair of coupled linear differential equations for
the sway and yaw velocities; they can be solved for prescribed rudder angles
if the hydrodynamic and body-mass coefficients are specified. Generally a
digital or analog computer must be used for this purpose, but a few qualitative conclusions can be drawn here without elaborate analysis. To simplify
the discussion as much as possible, we shall assume that the longitudinal
position of the origin is chosen such that
S
M 35
+ M 35 = 0.
(62)
For a ship in steady-state motion, with the initial conditions U3 = Ω2 =
0, the response to a sudden change of rudder angle can be approximated
by considering only the acceleration terms in (60–61). Thus for the initial
period of time before the sway and yaw velocities develop substantial values, it follows that
 1 πρT 2U 12 
U 3 =  2 S
δ R (t ),
 m33 + m 
(63)
 − 1 πρT 2U 12 xT 
Ω 2 =  2 S
δ R (t ).
 m53 + M 55 
(64)
364
Chapter 7
Figure 7.5
Trajectory of a ship in a steady-state turning maneuver.
Since xT is negative, both of the factors in parentheses are positive, and a
vessel commencing a turn to port will be accelerated initially to starboard,
as shown in figure 7.5.
The ultimate, steady-state turning motions can be analyzed by assuming
that U3 and Ω2 are constant. Thus, from (60) and (61),
1
2
πρT 2U 1δ R = mTU 3 − ( xT mT + m )Ω2 ,
S
− 21 πρT 2U 1δ R xT = −( m33
+ xT mT )U 3 + ( xT )2 mT Ω2 .
(65)
(66)
Using Cramer’s rule, the solutions of these coupled linear equations are
given by
U3 / δ R =
− 21 πρT 2U 1mxT
,
S
( mT xT ) − ( m + xT mT )( m33
+ xT mT )
(67)
Ω2 / δ R =
S
πρT 2U 1m33
.
S
( mT xT ) − ( m + xT mT )( m33
+ xT mT )
(68)
2
1
2
2
Since xT is negative, U3 has the same sign as Ω2, and a ship in a steady-state
turn is oriented with the bow pointed into the turn, as shown in figure 7.5.
This gives an effective angle of attack and a resulting inward force, to maintain the centripetal acceleration. The radius of the steady-state turn is equal
to U1/Ω2, to first order in the sway and yaw velocities.
The quantitative validity of (67–68) is limited because the denominator, or the determinant of (65–66), is close to zero for practical vessels. For
example, using the approximations indicated in the second column of
table 7.2, this determinant is equal to
Hydrodynamics of Slender Bodies
365
2
 1 πρlT 2   1 − BCB  ,
2
  2 πT 
where CB = ∀ / lBT is the block coefficient and B is the beam. For a ship with
block coefficient 0.7, this estimate of the determinant vanishes if B/T = 2.2.
For wider ships the determinant will be negative, and (68) predicts a turn
opposite to that associated with the normal action of the rudder. Under
these circumstances, where small changes in the values of the hydrodynamic forces affect the value of the determinant substantially, more accurate theoretical or experimental methods must be employed.
The vanishing of the determinant of (65–66) is related to the directional
stability of the ship, which can be analyzed in terms of the homogeneous
solutions of (60–61) with the rudder angle set equal to zero. Since (60–61)
are coupled first-order differential equations with constant coefficients, the
solutions for the sway and yaw velocities must be of the general form
U 3 = Re {c1eσ1t + c2 e σ 2t } ,
(69)
Ω2 = Re {c3eσ1t + c4eσ 2t } ,
(70)
where the constants cj and stability indices σj are complex.
Dynamic stability requires that the homogeneous solutions (69–70)
decay to zero exponentially with time, or that
Re(σ 1,2 ) < 0.
(71)
The resulting motion will then appear as shown in figures 7.6 (a—b). If
either or both indices violate this inequality, the vessel will be unstable, as
in figure 7.6 (c).
Quadratic equations for the stability indices can be obtained by substituting (69–70) in (60–61). With the rudder angle set equal to zero, the only
nontrivial solutions of (60–61) will occur when the determinant of the coefficients on the right side of these equations is zero. With the origin chosen
in accordance with (62), it follows that
S
U 1mT + ( m + m33
)σ j
−U 1( m + xT mT )
= 0.
S
S
−U 1( m33 + xT mT ) U 1( xT )2 mT + ( m55
+ M 55 )σ j
(72)
The resulting quadratic equation is of the form
Aσ 2j + Bσ j + C = 0,
(73)
366
Chapter 7
Figure 7.6
The three cases where the stability indices σi are real and negative (a), complex (b),
or real and positive (c). The left figures show typical positions of the stability indices
in the complex plane, and the right figures show the corresponding solutions for the
ship motions, as functions of time, subject to an initial disturbance from equilibrium. All scales are arbitrary.
where
S
S
A = ( m + m33
)( m55
+ M 55 ),
(74)
S
S
B = U 1mT [( m55
+ M 55 ) + ( xT )2 ( m + m33
)],
(75)
S
S
C = −U 12[ mm33
+ mT xT ( m33
+ m )].
(76)
The solutions of the quadratic equation (73) are given by
1
σ 1 
[ − B ± ( B2 − 4 AC )1/2 ].
 =
σ 2  2 A
(77)
Hydrodynamics of Slender Bodies
367
It is clear that (74–75) will be positive quantities, whereas the sign of (76) is
uncertain since xT is negative. It follows that
Re(σ 2 ) = Re
{
}
1
[ − B − ( B2 − 4 AC )1/2 ] < 0.
2A
(78)
However,
Re(σ 1 ) = Re
{
}
1
[ − B + ( B2 − 4 AC )1/2 ] < 0,
2A
> 0,
for C > 0,
(79)
for C < 0,
respectively. Thus C must be positive for stability, and the sign of (76) is
crucial. Essentially the same factor occurs in connection with the steadystate turning motion, as the denominator of (67–68), and thus stability in
a steady-state turn is necessary and sufficient to ensure dynamic stability in
the more general sense.
For a nonlifting body with a pointed tail, such that mT = 0, C is negative
and the vessel is always unstable. This situation results from the destabilizing effect of the Munk moment; in general an elongated nonlifting body
will be stable only when moving broadside to the flow. Directional stability
of a streamlined body in the longitudinal direction depends on a tail fin, as
in the case of an arrow or a wind vane. Specifically, directional stability of
a ship requires that
mT xT >
S
mm33
S
m + m33
(80)
For given values of the body mass and added-mass coefficients, the moment
arm xT must be sufficiently large for (80) to be satisfied. Noting our choice
of the origin in accordance with (62), we can achieve this objective by moving the center of gravity forward, as well as in the more obvious manner
by moving the tail fin aft. Thus the position of the center of gravity is of
great significance for the dynamic stability of ships, and indeed for other
vehicles as well1.
From the hydrodynamic standpoint, the stability criterion (80) can be
enhanced by increasing the added mass at the stern or by decreasing it elsewhere. Thus a ship’s stability will increase as the vessel is trimmed down at
the stern (see problem 7).
Excessive stability is undesirable in ship design because it impairs the
turning ability. Thus a small positive value for the parameter C is optimum.
368
Chapter 7
However, a limited degree of instability is often tolerated, particularly for
supertankers, with the premise that this can be controlled by a suitable
automatic pilot.
Since the sign of (76) is independent of the magnitude of U1, the forward velocity is not a factor in determining the directional stability of a
ship unless wave effects become significant, requiring a modification of the
hydrodynamic force coefficients in table 7.1. The stability of a submarine
in the vertical plane is speed-dependent, as noted in problems 8 and 9.
This distinction results because the submarine has a hydrostatic restoring
moment, independent of the forward velocity, and the relative importance
of this and the hydrodynamic forces varies with the forward velocity.
The characteristic time-scale for maneuvering is inversely proportional
to the stability indices; thus it is directly proportional to A/B or l/U1. For
constant Froude number, the time scale is proportional to l1/2. This result
is fundamental to the simulation of ship maneuvers for training purposes.
Human operators can sense the response of a 10 m model with much
greater ease than that of a 300m vessel. For this reason attempts to train
operators of supertankers with small free-running models have generally
been abandoned in favor of computer simulation, involving visual displays
and mock-up of the bridge controls, operated in the time-scale of the fullscale vessel.
Our analysis of ship maneuvering has been based on the slender-body
hydrodynamic force coefficients derived in section 7.4. This approach permits us to derive the most common form of the equations of motion from
rational fluid mechanics and to draw qualitatively correct conclusions. For
design purposes, greater accuracy is required; it is customary to assume the
same equations of motion, but to measure the force coefficients from experiments where a captive model is forced to oscillate sinusoidally.
The latter approach to this subject is discussed by Mandel (1967), Abkowitz (1969), and by several authors in Bishop, Parkinson, and Eatock Taylor
(1972). The survey by Norrbin (1970) deals with ship maneuvering in open
and restricted water. Frank et al. (1976) discuss the more general equations
of motion in convolution form, similar to (6.195), emphasizing the relation
between these and the simpler equations (52–53).
Hydrodynamics of Slender Bodies
369
7.6 Slender Bodies in Waves
The oscillatory motions of a slender body on the free surface can be
described in accordance with the linear theory developed in chapter 6, with
the additional assumption here that the body is slender. We shall consider
the separate cases of a vertical slender body, such as a spar buoy, and a horizontal slender body, such as a ship or submarine.
The body is defined by two disparate length scales, the length l and maximum lateral dimension d, with the slenderness parameter ε = d/l assumed
small. The presence of waves on the free surface introduces a third length
scale, the wavelength λ, and it is necessary to stipulate the order of magnitude of λ in relation to l and d. We shall consider explicitly the case where
the wavelength is comparable to the body length, λ/l = O(l), and thus the
body diameter is small compared to the wavelength. The converse problem
of short waves, with λ/l ≪ 1, will be treated in section 7.7.
The slender-body theory for a body in an infinite fluid can be generalized by imposing the linearized free-surface boundary condition (6.6). This
approach requires that the potential for a source beneath the free surface be
utilized. However, the leading-order forces acting on the body are independent of this complication and can be discussed in a relatively simple manner. Here we shall follow the latter course, guided by our earlier conclusions
regarding the orders of magnitude of the longitudinal and lateral flows and
of the resulting hydrodynamic forces on the body.
Proceeding as in section 6.15, we may express the velocity potential in
the form
6
 

φ = Re e iωt  ∑ ξ jφ j + A(φ0 + φ7 )  .

=
j
1

 
(81)
Here ξj denotes the complex amplitude of the six rigid-body motions, A is
amplitude of the incident wave, Aϕ0eiωt is the corresponding velocity potential, and Aϕ7eiωt is the scattered potential due to the disturbance of the incident wave by the body.
The incident wave potential is given by (6.16) or (6.23) for infinite or
finite water depth, respectively. Since this potential is independent of the
body, its order of magnitude with respect to ε is O(1). The remaining potentials ϕj (j = 1, 2, …, 7) are body potentials that will depend on ε.
370
Chapter 7
The body potentials can be analyzed in the inner region, near the body
section ∑B (x), using a two-dimensional approximation. The relevant length
scale is d = O(εl), and since ω2d/g = O(ε), the free-surface condition in the
inner problem can be replaced by a rigid boundary condition as in figure 6.21. The orders of magnitude of the various components of the body
potential, as well as the resulting force components acting on the body, can
be estimated as if the body and its image were situated in an infinite fluid.
Thus, in general, axial forces will be O(ε4 log ε), whereas lateral forces will
be O(ε2).
In addition to the hydrodynamic pressure forces associated with the
body potential, we must consider the forces due to the body mass, the
hydrostatic restoring force, and the Froude-Krylov component of the exciting force due to the pressure field of the incident wave. Thus, for each
mode of motion, four different types of force components are estimated, as
displayed in table 7.3.
For a vertical spar buoy, the estimates of the body-induced pressure
forces are shown in the first column. Heave induces a hydrodynamic force
of order (ε4 log ε), and one can infer from table 4.3 that the corresponding yaw moment will be proportional to d4, and hence O(ε4). The remaining motions are lateral with respect to the body axis, and induce forces of
order ε2.
The body-mass forces for a spar buoy can be estimated from (6.182–6.183),
and are as shown in the second column of table 7.3. The third column shows
the hydrostatic force and moment, with the heave force proportional to the
waterplane area O(ε2), and the roll and pitch moments proportional to the
moment of inertia of order ε4, as indicated by (6.144–6.149).
Finally, the Froude-Krylov component of the exciting force can be
estimated for a spar buoy from Gauss’ theorem, in a manner similar to
(6.165–6.166). The results are as shown in the fourth column of table 7.3.
In this case the displaced volume ∀ and waterplane area S are both O(ε2).
The contribution to the exciting force from the scattering potential ϕ7 is
included in the body-potential contributions listed in column one.
Similar estimates can be made for the forces on a horizontal body, as
shown in table 7.3. In surge, the body-induced force is longitudinal, and
hence O(ε4 log ε), while the body-mass force and the Froude-Krylov force
are O(ε2). For vertical motions in the inner region, associated with pitch
and heave, the two-dimensional flow is source-like since the submerged
Hydrodynamics of Slender Bodies
371
Table 7.3
Orders of Magnitude of the Different Types of Forces and Moments Acting on a
Vertical or Horizontal Slender Body, in Waves of Wavelength Comparable to the
Body Length
Vertical Body Axis
Mode
1. Surge
2. Heave
FB
FM
2
2
ε
ε
4. Roll
ε
2
5. Yaw
6. Pitch
2
ε
ε log ε
2
3. Sway
ε
4
2
ε
Horizontal Body Axis
FHS
0
2
ε
0
ε
2
ε
ε4
ε4
0
2
2
ε
ε
4
ε
4
FFK
FB
FM
FHS
FFK
2
4
2
0
ε2
2
ε
ε
2
0
ε2
ε
2
ε
2
ε log ε
2
ε log ε
ε
2
ε
ε
4
ε4
2
ε
2
ε
ε
ε
ε
ε
4
ε
ε2
ε2
ε2
0
ε2
2
2
ε
ε
ε log ε
ε
2
Note: FB denotes the body-induced pressure force, FM the body-mass force, FHS the
hydrostatic force, and FFK the Froude-Krylov force due to the pressure field of the
undisturbed incident wave. The dominant terms in each mode are indicated by
bold-face type (ε). In all cases the body length is assumed to be O(1).
sectional area of the body is changing in time. As a result the body potentials in these modes are O(ε log ε) and the corresponding force components
are O(ε2 log ε). The dominant vertical forces are the hydrostatic component,
proportional to the waterplane area of order ε, and the vertical component
of the Froude-Krylov force which, from (6.165), is of the same order. An
exception occurs for a submerged body, where the waterplane area is zero
and the remaining vertical force components are O(ε2).
Equations of motion for the six degrees of freedom can be derived from
the forces listed in table 7.3. In each mode the bold-faced terms are dominant, and thus the leading-order equations of motion are simplified. We
shall outline the details of this procedure for the vertical and horizontal
forces acting on a spar buoy, and for the vertical force acting on a slender
ship.
For a spar buoy situated at x = z = 0, the vertical Froude-Krylov force can
be expressed in the form
FFK = ∫∫ p0 ny dS SB
0
∫ S′( y ) p ( y ) dy.
0
(82)
−T
Here S(y) is the sectional area of the body, T is the draft, and p0(y) is the
pressure field of the incident wave system, at a depth y. Note that in
372
Chapter 7
analyzing the vertical force, the variation of the pressure p0 in the horizontal plane can be neglected, since the lateral dimensions of the body are
small compared to the wavelength. For deep water, using the incident wave
potential (6.16) and the linear form of Bernoulli’s equation (6.3), it follows
that
0


FFK = ρ gA Re e iωt ∫ S ′( y )e ky dy  .
 −T

(83)
Assuming the spar buoy is unrestrained, an equation of motion for
heave follows by equating the exciting force (83) to the body-mass and
hydrostatic forces. Thus
0
[ −ω 2 m + ρ gS( 0 )]ξ3 = ρ gA ∫ S ′( y )e ky dy ,
(84)
−T
where
0
m = ρ∀ = ρ ∫ S( y )dy
(85)
−T
is the body mass, and ξ3 is the complex heave amplitude. Equation (84) can
be solved for ξ3. An obvious feature of the result is an unbounded resonance
at the natural frequency
ω n = [ gS( 0 ) / ∀ ]1/2 .
(86)
Except for the immediate vicinity of resonance, the simple solution of
(84) is quite accurate, as shown in figure 2.16. To render this solution valid
near resonance, a damping coefficient of order ε4 can be derived from the
Haskind relations (6.173). However for long cylindrical buoys, the resulting
wave damping will be exponentially small in proportion to e−2kT. Under this
circumstance viscous damping may be significant. Another higher-order
effect that manifests itself near resonance, as shown in figure 2.16, is the
reduction of the natural frequency due to the added mass of the body.
A complementary analysis is required for the horizontal force on a vertical spar buoy and for the resulting motions of the buoy. The inner flow
near a body section can be analyzed using the technique developed in section 4.17, where the slowly varying velocity field is the horizontal component of the incident wave. Thus, if the incident waves are moving in the
Hydrodynamics of Slender Bodies
373
x-direction, the local force on a body section at a depth y is given from
(4.151) in the form
∂φ
Fx = −[ ρS( y ) + m11( y )]Re  −iω 0  − m11( y )U ( y )

∂x  x = 0
= [ ρS( y ) + m11( y )]Re(ω kφ0 )x = 0 − m11( y )U ( y ).
(87)
Here m11(y) is the local two-dimensional added-mass coefficient of the body
section, and U(y) is the body velocity in the horizontal direction. Substituting the incident wave potential (6.16) or (6.23) and integrating over the
body depth gives the total hydrodynamic force, which may be equated to
the body-mass force. A similar result follows for the pitch moment. For an
axisymmetric body in deep water, the details are carried out by Newman
(1963). In general the solutions for surge and pitch are coupled and they
are resonant unless the body is neutrally stable in pitch. Damping can be
inferred from the Haskind relations, noting that the term in (87) proportional to ϕ0 is the exciting force.
If the buoy diameter is comparable to or smaller than the wave amplitude, cross-flow viscous drag effects will become important, as discussed in
section 2.14. To account for this in a semiempirical manner, an additional
force of the following form may be added to (87),
1
∂φ
∂φ 0
ρCDl 2  0 − U 
−U ,
 ∂x
 ∂x
2
as in equation (2.49). Here CD is a viscous drag coefficient, for lateral flow
across the body section, and l2 is the area upon which the drag coefficient
is based.
Turning our attention to the case of a horizontal floating body, the
vertical force is dominated by the terms of order ε in table 7.3, that is,
the hydrostatic restoring force and the Froude-Krylov exciting force. The
Froude-Krylov force can be evaluated using (6.160) and (6.165), in the
form
FFK
′ = ρ g Re { AB( x)e i(ωt − kx cosθ ) } ,
(88)
where B(x) is the local beam of the waterplane and θ is the angle of incidence as shown in figure 6.6. Simple equations of motion follow for the
coupled heave and pitch motions in the form
374
Chapter 7
Sξ2 + S1ξ6 = A∫ B( x)e − ikx cosθ dx,
(89)
S1ξ2 + S11ξ6 = A∫ B( x)e − ikx cosθ x dx.
(90)
l
l
Here the terms on the left-hand side result from the hydrostatic restoring
force and moment (6.144–6.149), with S the waterplane area, and S1 and S11
the first and second moments of this area.
Inertial and hydrodynamic effects are of higher order as indicated in
table 7.3, and thus are absent from (89–90). As a result, the solutions of
these equations are restricted to the range of frequencies substantially less
than the resonant frequencies of the ship. The resonant frequencies ωn can
be estimated, as in (86), by equating the hydrostatic and virtual-mass forces
which are of order ε and ε2ω2 respectively. It follows that ωn = O(ε−1/2), and in
deep water the resonant wavelength will be of the order of the ship’s beam.
Thus resonant ship motions must be analyzed from the complementary
short-wavelength theory where λ/d = O(1).
Typical solutions of (89–90) are shown in figures 7.7–7.8 for an aircraft carrier in head seas (θ = π). Also shown are experimental results for
zero forward speed and a Froude number of 0.14. At zero speed the agreement is reasonable, except for an apparent resonance near λ/l = 0.5. The
importance of this resonance is diminished because the exciting force and
moment are small for λ < l. However if the ship proceeds with substantial
forward velocity in head seas, the frequency of encounter is increased by
the Doppler shift. Thus, the oscillation frequency is increased for a given
wavelength, ultimately to the point where resonant motions occur in the
range of substantial exciting forces and moments. This situation is apparent
from the experiments shown in figures 7.7–7.8, and would become progressively more serious at higher speeds. In this case the practical value of the
long-wavelength slender-body theory is diminished, and a complementary
approach must be developed where the magnitude of the oscillation frequency is in the range of resonant motions, of order ε−1/2.
7.7 Strip Theory for Ship Motions
The problem of a slender ship moving with forward velocity in a seaway
is one of the most important topics in ship hydrodynamics. Here, by
Hydrodynamics of Slender Bodies
375
Figure 7.7
Heave motions predicted from (89–90) and compared with experiments. The wave
frequency ω0 is defined with reference to the fixed coordinate system, as in (91).
(Adapted from Newman and Tuck 1964)
376
Chapter 7
Figure 7.8
Pitch motions predicted from (89–90) and compared with experiments. (Adapted
from Newman and Tuck 1964)
Hydrodynamics of Slender Bodies
377
comparison to the simpler problem of a slender body in waves without forward speed, hydrodynamic interactions will exist between the steady-state
flow field due to the ship’s forward motions and the oscillatory flow associated with the unsteady motions. Since complete analysis of these interactions is beyond the present state of knowledge in this field, we must adopt
a pragmatic approach, justified by experimental confirmation.
Two important effects of the ship’s forward velocity can be analyzed
without major difficulty. One is the change in the frequency of encounter,
which is an elementary feature of the ship-motion problem but one with
far-reaching consequences. The second effect is the longitudinal convection of momentum, which changes the oscillatory force distribution along
the hull, as in the simpler case of a slender body in an unbounded fluid.
The effect of the forward speed on the frequency of encounter is identical to the Doppler shift in other fields of wave motion. Restricting our
attention to harmonic motions in deep water, the velocity potential of
the incident wave system given by (6.16) can be written in the complex
form
φ = (igA / ω 0 ) exp [ ky 0 − ik( x0 cos θ + z0 sin θ ) + iω 0t ],
(91)
where the real part is implied wherever an exponential time-dependence
is displayed. Here A is the amplitude of the incident wave, θ is the direction of propagation relative to the x-axis (see figure 6.6), ω0 is the wave
frequency with respect to a fixed reference frame (x0, y0, z0), and k = ω 02 / g
is the wavenumber.
In a reference frame (x, y, z) moving with the steady forward motion of
a ship, the transformations (2–4) apply if the ship moves in the positive
x-direction. Thus, (91) can be expressed in the form
φ = (igA / ω 0 )exp[ ky − ik( x cos θ + z sin θ ) + i(ω 0 − kU cos θ )t ].
(92)
The wave amplitude A and wavenumber k are unchanged from (91), but the
frequency of encounter in the moving reference frame is given by
ω = ω 0 − kU cos θ .
(93)
For following waves (θ = 0), ω < ω0, but in head waves (θ = π), the converse is
true. The same conclusions apply to a lesser extent for quartering (0 < θ < π/2)
and bow (π/2 < θ < π) waves, whereas for beam waves (θ = π/2), the two frequencies coincide.
378
Chapter 7
The pressure field of the incident wave system (91–92) can be derived in
either reference frame. Using the linearized form of the Bernoulli equation
(6.3), it follows from (91) that
p = −ρ
∂φ
= − ρ(iω 0 )φ = ρ gA exp[ ky 0 − ik( x0 cos θ + z0 sin θ ) + iω 0t ].
∂t
(94)
Here the hydrostatic component of the pressure is neglected because this
is not affected by the incident waves. In the moving reference frame, the
time-derivative must be transformed, in a manner analogous to (19); thus
∂φ
∂φ
p = −ρ 
−U 
 ∂t
∂x 
= − ρ(iω + ikU cos θ )φ
= ρ gA exp[ ky − ik( x cos θ + z sin θ ) + iωt ].
(95)
The results of (95) are identical to (94), except for the change in frequency
of encounter. Thus the incident-wave pressure field is not affected by the
choice of reference frame.
We shall restrict our attention to the heave and pitch motions of a ship,
in incident waves of the form (92). As noted at the end of section 7.6, the
resonant frequencies in these modes are of order ε−1/2, implying that the
frequency of encounter ω is of the same order of magnitude in the regime
where significant motions occur. For ω to be large relative to the wave frequency ω0, it follows that the incident wave angles of greatest importance
are head waves and bow waves. Thus we shall consider this regime, with the
assumption that ω = O(ε–1/2) or, in terms of the ship’s beam B, ω2B/g = O(1).
As in section 6.15, the results are linearized in terms of the small oscillatory
amplitudes of the incident wave and resulting body motions. On this basis,
added-mass and damping coefficients can be derived for forced motions of
the ship in otherwise calm water, and the exciting force and moment can
be analyzed with the heave and pitch oscillations suppressed. In both problems the Froude number U/(gl)1/2 is assumed to be O(1).
First we consider the forced heave and pitch motions in calm water, in
the frequency regime where ω2B/g = O(1). Under this circumstance, and
unlike the low-frequency regime of section 7.6, both terms must be retained
in the linearized free-surface condition (6.6). The motion is periodic in the
moving reference frame, where (6.6) takes the form
Hydrodynamics of Slender Bodies
2
 ∂ − U ∂  φ + g ∂φ = 0
 ∂t
∂x 
∂y
379
(96)
or
−ω 2φ − 2iωU
∂φ
∂ 2φ
∂φ
= 0, on y = 0.
+U2 2 + g
∂x
∂x
∂y
(97)
The total vertical displacement of the ship, at a longitudinal position x,
is given by
η( x, t ) = (ξ2 + xξ6 )e iωt ,
(98)
where ξ2 and ξ6 are the complex amplitudes of heave and pitch, as defined
in figure 6.19. This vertical motion, as well as the ship’s geometry, are
slowly varying in the x-direction. Thus a two-dimensional approximation
equivalent to (16) can be adopted for the velocity potential in the inner
region, near the ship hull, in the form
φ = U Φ1 + V Φ 2 .
(99)
Here U is the forward velocity, and
∂
∂
V ( x, t ) =  − U  [ e iωt (ξ2 + xξ6 )]
 ∂t
∂x 
(100)
is the local vertical velocity of the ship hull with respect to the fixed reference system.
Equation (100) is analogous to (19), for the lateral velocity of a slender
body. Since ξ2 and ξ6 are constant, the only contribution from the term
U∂/∂x is an apparent vertical velocity equal to the product of the pitch angle
and the forward speed.
The velocity potentials Φ1 and Φ2 in (99) satisfy the free-surface condition (6.6) with respect to the fixed reference frame, or (97) with respect to
the moving reference frame, as well as the two-dimensional Laplace equation (12) and a suitable radiation condition.2 On the hull surface, Φ1 satisfies the boundary condition (17), whereas
∂Φ 2
= ny ,
∂N
on Σ B ( x).
(101)
From the slenderness assumption, and the estimates (10), it can be anticipated that the potential Φ1 will be of order ε2, or possibly O(ε2 log ε), as in
380
Chapter 7
an infinite fluid; but the potential Φ2 associated with the vertical motion
will be of order ε. Thus, in terms of the slenderness parameter, the potential
Φ2 is dominant.
The differential vertical force acting on the ship can be derived as in
(36), in a reference frame fixed with respect to the undisturbed fluid. The
control contour surrounding the hull must include the portion of the free
surface, ∑F, between ∑B (x) and the contour ∑∞ at large distance from the
body. Thus
Fy′ = − ρ
d
dt
∫
φny dl − ρ
ΣB ( x)
∂
∂x0
∂φ ∂φ
dy 0 dz0
0 ∂y 0
∫∫ ∂x
S0
1
 ∂φ ∂φ

−ρ ∫ 
− ny ∇φ ⋅ ∇φ  dl ,


y
n
2
∂
∂
0
Σ F + Σ∞
(102)
where S0 is the portion of the plane x0 = constant interior to the closed contour ∑B (x) + ∑F + ∑∞.
If we restrict our attention to the linearized force, of first-order in the
unsteady body motions and of leading order in the slenderness parameter
ε, the nonlinear terms in (102) may be ignored. Thus, the local vertical force
is given by the expression
Fy′ = − ρ
= −ρ
d
dt
∫
φny dl
ΣB ( x)
d
[V ( x, t ) ∫ Φ 2 ny dl ].
dt
ΣB ( x)
(103)
The integral in (103) can be treated in a manner analogous to (38). Since
the wave motion is periodic with respect to the moving coordinate system,
the potential Φ2(y, z; x) will be complex but independent of time. Thus the
contour integral in (103) can be defined in terms of generalized added-mass
and damping coefficients for the two-dimensional section,
ρ
∫
Φ 2 ny dl ≡ a22 ( x,ω ) +
ΣB ( x)
1
b22 ( x,ω ) ≡ Z( x,ω ).
iω
(104)
The desired expression for the differential vertical force follows by substituting (104) in (103),
d
[V ( x, t )Z( x,ω )]
dt
∂
∂
= −  − U  [V ( x, t )Z( x,ω )].
 ∂t
∂x 
Fy′ = −
(105)
Hydrodynamics of Slender Bodies
381
This expression differs from (39) only with respect to the change in direction of the body motion, and the substitution of the complex coefficient
(104) for the added-mass.
The total heave force is given by the integral of (105). With the assumption that (104) vanishes at the bow and stern, it follows that
U
Fy = ω 2 e iωt ∫ ξ 2 +  x −  ξ6  Z( x,ω )dx.

ω 
i

l
(106)
Similarly, after a partial integration of (105), the pitch moment is obtained
in the form
U
U2 


M z = ω 2 e iωt ∫ ξ2  x +  + ξ6  x2 + 2   Z( x,ω ) dx.


ω 
iω 
l 
(107)
The factors of the pitch and heave amplitudes, in (106) and (107), are the
complex force coefficients fij defined in (6.150). The three-dimensional
added-mass and damping coefficients follow by taking the real and imaginary parts of these factors.
In this manner the three-dimensional force coefficients are derived from
a strip-theory synthesis, as integrals of the two-dimensional force coefficient (104). However, the forward speed U leads to additional components
of the force and moment (106–107) which are not evident from a simple
integration of (104) along the length. These additional contributions result
in part from momentum convection, as well as from the speed-dependent
term in the vertical body-velocity (100).
The coefficients a22 and b22 in (104) are closely related to the twodimensional added-mass and damping coefficients for forced motion of a
two-dimensional body, with profile ∑B, oscillating on the free surface with
frequency ω. The only difference between these problems is in the freesurface boundary condition. The extra terms in (97), depending on U, distinguish the potential Φ2 from the two-dimensional solutions described in
chapter 6. Physically, these imply an interaction between adjacent sections
of the ship hull, depending on the rate of change in the longitudinal direction of the inner solution.
The only role of the x-coordinate on Φ2 is the parametric dependence on
the hull profile ∑B(x). Thus Φ2 will be purely two dimensional in those cases
where ∑B is locally independent of x, as along the parallel middle body of a
ship hull. In this particular case the speed-dependent terms in (97) vanish,
382
Chapter 7
and the force coefficients a22 and b22 in (104) are identical to the added-mass
and damping coefficients for the two-dimensional cylinder, with profile ∑B.
For this particular case results such as those illustrated in figure 6.23 can be
substituted for the complex force coefficient Z and used to compute the differential force (105), as well as the total force and moment (106–107). Thus,
for vertical heaving motions of a cylindrical body with constant profile,
the differential force (105) is independent of the forward speed U. This is
confirmed physically since a long cylindrical body can be translated in the
axial direction, in an inviscid fluid, without affecting the surrounding flow
except near the ends.
In the general case of a ship hull with varying profile ∑B, the difficulties associated with the speed-dependent terms in (97) can be avoided by
noting that U/ω = O(ε1/2). Thus, to leading order, (97) can be replaced by
the linearized free-surface condition (6.126) for sinusoidal motion without forward speed. Once again, the zero-speed two-dimensional results can
be used for the added-mass and damping coefficients in (104). Salvesen,
Tuck, and Faltinsen (1970) proceed in this manner, using expressions
equivalent to (106–107). Experimental confirmation is provided by Vugts
(1970), for both the total force and moment (106–107) and the local force
distribution (105).
The approach outlined is deficient from the theoretical standpoint, since
some higher-order terms proportional to U/ω = O(ε1/2) have been retained,
while others have been neglected. Indeed, one term retained in (107) is
U2/ω2 = O(ε). In a consistent leading-order theory these terms should be
discarded. Alternatively, a consistent higher-order theory should be developed, where all terms of the same order are retained in the free-surface
condition (97) and in the integral over ∑F in (102).
Ogilvie and Tuck (1969) have performed a consistent analysis, to order
ε1/2, including the two additional effects and a nonlinear contribution
to the free-surface boundary condition. These complicated higher-order
effects largely cancel each other, and to order εl/2 the total force due to
heave and the moment due to pitch are correctly given by (106) and (107)
respectively.
The analysis of Ogilvie and Tuck (1969) does reveal additional contributions to the cross-coupling coefficients, for the heave force due to pitch
and the pitch moment due to heave, involving integrals of (Φ2)2 over the
free surface. These are re-derived from an energy analysis by Wang (1976).
Hydrodynamics of Slender Bodies
383
Computations have been carried out by Faltinsen (1974), who shows
improved agreement of the resulting cross-coupling coefficients with experiments. The net effect of these additional terms on the pitch and heave
motions in waves is not well established.
In essence, the results (105–107) with the zero-speed two-dimensional
added-mass and damping coefficients are relatively simple to use and give
useful engineering predictions. The more complete results of Ogilvie and
Tuck (1969) are superior from the mathematical standpoint, but their practical value is open to debate. The simple approach includes some forwardspeed effects, which are identical to those associated with the lateral force
acting on a slender body in an unbounded fluid. The additional forwardspeed effects included by Ogilvie and Tuck (1969) result from the free surface. Apparently the latter are relatively unimportant, but this hypothesis
lacks a rational justification.
The exciting force and moment can be analyzed with a similar approach,
but under more restrictive assumptions. Assuming the forward speed U =
O(l), with respect to ε, and the frequency of encounter ω = O(ε−1/2), the only
possible estimates for ω0 and k compatible with (93) and the dispersion relation k = ω 02 / g are k = O(ε−1/2) and ω0 = O(ε−1/4). Thus the incident wavelength
2π/k must be short compared to the ship length, but long compared to the
beam.
The exciting force and moment are determined from the solution of the
diffraction problem, where the oscillatory potential consists of the incident
wave ϕ0, plus a scattering potential ϕ7, as in (6.123). The contributions to
the exciting force from these two potentials can be treated separately. The
Froude-Krylov force, associated with the pressure field of the incident wave
system, can be obtained directly from (95) in the form
Fy′0 = ρ gA exp( −ikx cos θ + iωt )
∫
exp( ky − ikz sin θ )ny dl.
(108)
ΣB ( x)
This integral can be evaluated numerically, for a prescribed section ∑B (x),
and no further approximation of the Froude-Krylov force is necessary from
the computational standpoint. However, since k = O(ε−1/2), it follows that
(ky, kz) = O(ε1/2) on ∑B. Thus it is consistent with assumptions to be made in
deriving the total exciting force to expand the integral of (108) in a Taylor
series. Integrating the leading terms of this expansion gives
384
Chapter 7
Fy′0 ρ gA exp( −ikx cos θ + iωt )
∫
(1 + ky − ikz sin θ )ny dl
ΣB ( x)
(109)
= ρ gA exp( −ikx cos θ + iωt )[ B( x) − kS( x)],
where B(x) and S(x) are the local waterplane beam and sectional area, respectively, and symmetry about z = 0 has been assumed. The error incurred in
approximating (108) by (109) can be computed for simple sectional shapes.
For a rectangular section, (109) underestimates the Froude-Krylov exciting
force by 30 percent when the wavelength/draft ratio is 10. For sections with
less fullness, the error is smaller (see problem 11).
The scattering potential ϕ7 can be analyzed in the inner region with the
same two-dimensional approach used for the forced-motion problem. In
a reference frame moving with the ship, the boundary condition for ϕ7 is
(6.124). Substituting the incident wave potential (92), it follows that
∂φ7
∂φ
=− 0
∂n
∂n
= −iω 0 ( ny − inx cos θ − inz sin θ )
(110)
× exp[ ky − ik( x cos θ + z sin θ ) + iωt ].
Neglecting the normal component nx = O(ε), and recalling that (ky, kz) =
O(ε1/2) on the body surface,
∂φ7
−iω 0 ( ny − inz sin θ )exp( −ikx cos θ + iωt ).
∂n
(111)
The form of this boundary condition suggests that the scattering potential can be expressed in the inner region by
φ7 = −iω 0 (Ψ 2 − i sin θΨ 3 )exp( −ikx cos θ + iωt ),
(112)
where the functions Ψ2 and Ψ3 satisfy the boundary conditions
∂Ψ 2
= ny ,
∂N
(113)
∂Ψ 3
= nz ,
∂N
(114)
on the body contour ∑B. From (113–114) it can be inferred that Ψ2 and Ψ3
are slowly varying functions of the longitudinal position along the ship
length, in the same sense as the corresponding forced-motion potentials
Φ2 and Φ3.
Hydrodynamics of Slender Bodies
385
Operating with Laplace’s equation on (112), and neglecting longitudinal derivatives of Ψ2 and Ψ3, it follows that these are solutions of the
two-dimensional Laplace equation (12) to leading order in kB. However,
application of the free-surface condition (96–97) to (112) gives the boundary condition
−ω 0 2Ψ j − 2iω 0U
∂Ψ j
∂2Ψ j
∂Ψ j
+U2
+g
= 0,
2
∂x
∂x
∂y
on y = 0 ( j = 2, 3).
(115)
The only difference between the boundary-value problems for the functions Ψj and the corresponding forced-motion potentials Φj is that the
frequency of encounter ω appears in (97), whereas the incident wave frequency ω0 appears in (115). Thus the potentials Ψj can be regarded as forcedmotion potentials, with oscillation occurring at the incident wave frequency ω0, as
measured in a fixed reference frame.
Restricting our attention to the vertical exciting force and moment, for
a ship with symmetry about the plane z = 0, the potential Ψ3 in (112) can
be neglected. Substituting (112) in (103), the contribution from ϕ7 to the
differential force is given by
∂
∂
F y′ 7 = − ρ  − U  ∫ φ7ny dl
 ∂t
∂x  Σ B ( x )

∂
∂ 
= ρω 0 A  − U  i exp( −ikx co
os θ + iωt ) ∫ Ψ 2 ny dl 
 ∂t
∂x  

ΣB ( x)
∂
∂

= ω0 A
[ i exp( −ikx cos θ + iωt )Z( x,ω 0 )].
−U
 ∂t
∂x 
(116)
Here (104) has been used, along with the conclusion following (115).
Adding the forces (109) and (l16), the differential exciting force is
obtained in the form
Fyex
′ = A{ρ g[ B( x) − kS( x)]exp( −ikx cos θ + iωt )
∂
∂
+ iω 0  − U  [ z( x,ω 0 )exp( −ikx cos θ ) + iωt ]}.
 ∂t
∂x 
(117)
Before integrating (117) along the ship length, we note the special case
of a cylindrical body with profile ∑B independent of x, such as the parallel
middle body of a ship, or a horizontal pipeline. In this case B, S, and Z are
constants, and (l17) reduces to the expression
386
Chapter 7
Fyex
′ = gA{ρ B − k[ ρS + Z(ω 0 )]}exp( −ikx cos θ + iωt ).
(118)
Here the only effect of the forward speed is on the frequency of encounter,
for fixed wave amplitude A and wavenumber k. This result is consistent
with the fact, noted earlier, that axial motion of a slender cylindrical body
in an inviscid fluid has no effect on the inner flow except near the ends.
The steady forward velocity of a cylindrical body through a fluid is identical to the flow of a current past a fixed body. Thus (118) can be applied in
the case where a floating or submerged pipeline is fixed in position, in the
presence of waves and a superposed axial current. From this relation it follows that such a current has no effect on the wave force, provided the wave
amplitude and wavenumber are correctly measured in a frame of reference
moving with the fluid.
The heave exciting force on a ship is given by the integral of (117) along
the length, in the form
Fyex = gAe iωt ∫ e ikx cosθ {ρ B( x) − ρkS( x)
l
− k [1 − (ω 0U / g )cos θ ] Z( x,ω 0 )}dx,
(119)
where (93) has been used, and it is assumed that the factor in braces vanishes at the bow and stern. Here we note that the Froude-Krylov force, associated with the first two terms in braces, is independent of the forward
velocity, whereas the remaining contribution due to the scattering potential is a linear function of the forward velocity.
The pitch exciting moment can be derived in a similar manner, after
integrating by parts, with the result
M zex = gAe iωt ∫ e − ikx cosθ {ρ[ B( x ) − kS( x )]x
l
− [(1 − (ω 0U / g )cos θ ) kx − i(ω 0U / g )] Z( x,ω 0 )}dx.
(120)
Here again the Froude-Krylov component is independent of the forward
speed, while the scattering component is linear in U.
A complementary derivation of the exciting force (119) and moment
(120) is given by Salvesen, Tuck, and Faltinsen (1970), assuming that k =
O(ε−1) and using a generalization of the Haskind relations (6.172). In that
approach the forced-motion potential Φ2 enters as a consequence of the
Haskind relations, and as a function of the frequency of encounter instead of
the wave frequency ω0. An alternative approach based on using the Haskind
Hydrodynamics of Slender Bodies
387
relations in a two-dimensional manner is outlined by Newman (1970), and
is more closely related to the present results. The Haskind relations lead
to an exponential factor in the scattering-force integral of (116), in a form
similar to the Froude-Krylov integral (108). This additional factor makes the
scattering force more difficult to compute, since the force due to the scattering potential cannot be related directly3 to the added-mass and damping
coefficients.
These ambiguities are not the only uncertainties in the exciting force
problem. In general, oscillatory integrals such as (119) and (120) will tend
to cancel along the length of the ship, if the wavelength is short compared
to the ship length, as assumed here. Under these circumstances such integrals are dominated by local behavior near the ends. The importance of end
effects is emphasized by Faltinsen (1973) in a study of the exciting force for
the case where the incident wavelength is comparable to the ship’s beam
and the forward velocity is small.
The strip theory for predicting heave and pitch is completed by adding the body-mass and hydrostatic forces as given by (6.182–6.183) and
(6.144–6.149), respectively. The result is a pair of coupled linear equations
for the complex amplitudes of heave and pitch, analogous to (89–90). In
solving these equations of motion the principal task is the computation of
the two-dimensional added-mass and damping coefficients for each section
of the ship.
Typical results are shown in figure 7.9 for a “Mariner” class ship. The
calculations here are based on the exciting force and moment (119–120).
Results using the complementary derivation of the exciting force and
moment, given by Salvesen, Tuck, and Faltinsen (1970) and by Vugts
(1970), show similar agreement with experiments.
It is apparent from figure 7.9 that the pitch and heave motions in head
waves are predicted with sufficient accuracy for most practical purposes.
Moreover, the regime of agreement with experiments is not limited to high
frequencies in the vicinity of resonance, since the strip theory predicts the
correct (hydrostatic) low-frequency limit of the exciting force and moment.
Thus, as ω0 → 0, ξ2/A → 1 and |ξ6|l/(2A) ≃ ω02l/(2g), in agreement with the
results shown in figures 7.7 and 7.8. As a result, the strip theory is useful
over the complete range of frequencies of practical significance.
388
Chapter 7
Figure 7.9
Amplitude and phase of the heave and pitch motions for a “Mariner” ship in head
waves, at a Froude number of 0.3. The curves are strip-theory calculations. Experimental points are from Salvesen and Smith (1971). Amplitudes are nondimensionalized as in figures 7.7 and 7.8. The phase lead of the motion is shown, relative to the
incident wave elevation at the midship section. (Calculations courtesy of C. M. Lee)
7.8 Slender Bodies in Shallow Water
Slender-body theory can be applied to the study of ship hydrodynamics in
shallow water, and thus to the prediction of shallow-water effects on the
hydrodynamic characteristics of ships. These predictions are of particular
importance for large, deep-draft vessels during operations in coastal areas
and harbors.
With the same assumptions regarding the body geometry as in earlier
sections of this chapter, we shall assume in addition that the fluid is of
constant depth h = O(ε). This order of magnitude is assumed, so that the
depth appears in the inner problem. In the converse case, where the depth
is of the order of the body length, there are no effects of the bottom in the
inner region.
The outer flow, far from the inner region near the body surface, is characterized by waves on the free surface. Since the depth is small, these waves
will be nondispersive, as noted following (6.28). In order to describe this
Hydrodynamics of Slender Bodies
389
relatively simple wave motion, we shall assume that the characteristic
wavelength is O(1), and thus kh = O(ε), where k is the wavenumber 2π/λ.
With this restriction, the three-dimensional velocity potential in the outer
region can be expanded in a Taylor series about the bottom, of the form
 ∂φ 
φ( x0 , y 0 , z0 , t ) = φ( x0 , − h, z0 , t ) + ( y 0 + h ) 
 ∂y 0  y0 =− h
 ∂ 2φ 
+ ( y0 + h)  2 
+ .
 ∂y 0  y0 =− h
1
2
(121)
2
The first derivative vanishes on the bottom, and hence, to second order, the
potential can be expressed as
φ φ ( 0 ) ( x0 , z0 , t ) + 21 ( y 0 + h )2 φ ( 2 ) ( x0 , z0 , t ).
(122)
Evaluating the first and second derivatives of (122) with respect to y0, and
substituting these in the free-surface condition and Laplace’s equation,
respectively,
∂ 2φ ( 0 )
+ ghφ ( 2 ) = 0,
∂t 2
(123)
∂ 2φ ( 0 ) ∂ 2φ ( 0 )
+ φ ( 2 ) = 0.
+
∂x02
∂z02
(124)
Note that (123) holds on y0 = 0, and (124) holds throughout the fluid
domain (−h < y0 < 0), but since the functions ϕ(0) and ϕ(2) are independent of
y0, that distinction is not significant. Thus (123–124) are valid throughout
the x0-z0 plane in the domain of the outer solution.
Combining (123–124) to eliminate ϕ(2) gives the linearized equation for
shallow-water waves in the form
∂ 2φ ( 0 ) ∂ 2φ ( 0 ) 1 ∂ 2φ ( 0 )
+
−
= 0.
gh ∂t 2
∂x02
∂z02
(125)
This is the wave equation, which governs two-dimensional acoustic waves
with the speed of sound c replacing (gh)1/2 in (125). Thus shallow-water
waves are closely related to acoustic waves.
For steady-state motion in a moving reference frame, the time derivative
∂ /∂t is replaced by −U∂/∂x, and (125) takes the form
(1 − Fh2 )
∂ 2φ ( 0 ) ∂ 2φ ( 0 )
+
= 0,
∂x 2
∂z 2
(126)
390
Chapter 7
where Fh is the Froude number based on the fluid depth,
Fh = U / ( gh )1/2 .
(127)
With this parameter replaced by the Mach number U/c, (126) is the equation governing linearized steady-state two-dimensional flow in aerodynamics, with compressibility effects represented by the speed of sound.
From the mathematical standpoint, (126) is an elliptic equation in the
subcritical case (Fh < 1), and a hyperbolic equation in the supercritical case
(Fh > l). Physically these correspond respectively to the aerodynamic conditions of subsonic and supersonic flow. From either viewpoint the flow
is expected to change markedly as one passes through the critical Froude
number Fh = 1, and separate solutions of (126) must be developed for each
case.
In the subcritical case, (126) can be reduced to Laplace’s equation by the
Prandtl-Glauert transformation. If the lateral coordinate is transformed by a
factor
z ′ = z(1 − Fh2 )1/2 ,
(128)
while the longitudinal coordinate x = x′ is unchanged, it follows that
∂ 2φ ( 0 ) ∂ 2φ ( 0 )
+
= 0.
∂x ′ 2
∂z ′ 2
(129)
For supercritical flow, Fh > 1, the method of characteristics can be used to
show that the general solution of (126) is of the form4
φ ( 0 ) ( x, z ) = f [ x ± ( Fh2 − 1)1/2 z ].
(130)
For steady forward motion in the positive x-direction, the indeterminacy of
the sign in (130) is removed by imposing a radiation condition. Thus only
the characteristics that radiate downstream from the body are retained, and
if the motion is symmetrical about z = 0 it follows that the velocity potential is of the form
φ ( 0 ) ( x, z ) = f [ x + ( Fh2 − 1)1/2 z ].
(131)
In the inner region, near the body surface, motions that are slowly varying in the longitudinal direction can be approximated as two-dimensional
flows. If we restrict our attention to the low-frequency regime, as in section 7.6, the inner solution is governed by the rigid free surface condition
Hydrodynamics of Slender Bodies
391
∂ϕ/∂y = 0 on y = 0, and the same condition on the bottom y = h. Thus the
inner flow is constrained by two parallel and closely spaced rigid boundaries, as opposed to the unbounded case where the flow can spread radially
in the y-z plane. This constraint is an essential feature of the inner flow and
one of the principal effects of shallow water.
For lateral motions, the leading-order results for small ε can be constructed from a strip-theory synthesis, as in deep water (see problem 14).
Thus, the sway added mass of a body in shallow water can be represented
as in (40), but the relevant two-dimensional added-mass coefficient m22(x)
is determined for the body section ∑B (x) and its rigid-body image, in a
channel of total depth 2h. Some examples of the resulting two-dimensional
added-mass coefficient are discussed in section 4.18. Typically, as in (4.157),
the lateral added-mass coefficient is increased by proximity to the bottom.
The lateral maneuvers of ships in shallow water can be described as in
sections 7.4 and 7.5 provided the two-dimensional added-mass coefficient
is modified to account for shallow-water effects. As a result, ships are generally more stable in shallow water and more difficult to maneuver.
Lateral flows that can be described in the inner region without influence
from the outer flow are not affected to leading order by wave effects or by
the Froude number Fh. However, the strip-theory approach for lateral flows
breaks down when the “gap” between the bottom of the fluid and the body
is very small, resulting in a “blockage” of the lateral flow past the body section. In the limit when this gap tends to zero (or the body runs aground!),
the lateral flow must pass around the body ends, in a manner that is predominantly two-dimensional in the horizontal plane x-z, rather than in the
lateral plane y-z.
The transition between these two regimes can be treated by a matching
technique in which the outer flow is horizontal. Here, the extent to which
the inner flow passes through the gap or around the body ends depends
on the pressure jump between the two sides of the body, which depends
in turn on the inner limit of the outer solution. This technique has been
developed for the case Fh = 0 by Newman (1969) and extended to subcritical
Froude numbers 0 < Fh < 1 by Breslin (1972). The same technique is applied
to horizontal motions in a seaway by Beck and Tuck (1972).
For longitudinal flows, as in the case of an unbounded fluid treated
in section 7.2, it is essential to match the source-like flow in the inner
region with an outer three-dimensional solution. This matching procedure
392
Chapter 7
determines an arbitrary function f(x, t) in the inner solution, which affects
the pressure distribution and resulting forces on the body. We shall illustrate this situation for the steady forward motion of a ship in shallow
water, a problem that was solved first, in the sense described here, by Tuck
(1966).
The problem to be solved is that of the steady forward motion of a slender ship moving on the free surface in shallow water. Since the motion is
steady state in the moving reference frame (x, y, z), the inner solution near
the hull surface can be expressed in the form
φ( x, y , z ) = U [Φ1( y , z ; x) + f ( x)].
(132)
Here Φ1 is a solution of the two-dimensional Laplace equation (12), and as
in (17), the boundary condition
∂Φ 1
= nx , on Σ B ( x).
∂N
(133)
In addition, ∂Φ1/∂y = 0 on the free surface y = 0 and on the bottom y = h.
The function f(x) in (132) is a homogeneous solution of this boundaryvalue problem. As such, f(x) is indeterminate from the inner solution,
and must be found by matching (132) with the outer solution, where
three-dimensional effects are significant but where the body geometry is
unimportant.
Because of the boundary condition (133) and the relation (21), the solution for Φ1 is source-like with a net flux Q equal to (minus) the rate of
change of sectional area S′. In shallow water this flux is constrained by the
free surface y = 0 and the bottom at y = −h. Thus, far from the body in the
inner region, the flux must be a horizontal streaming flow, divided symmetrically in the two directions. The velocity of this streaming flow is Q/2h,
and the outer expansion of the inner solution must be of the form
∂Φ 1
∓ S ′ / 2h,
∂z
for z / ε → ± ∞.
(134)
The decomposition (132) can be made unique by integrating (134),and
specifying the asymptotic form of Φ1 as
Φ1 −( S ′ / 2h ) z , for z ε .
(135)
Equations (134–135) hold in the overlap region ε ≪ y ≪ 1, where they
are to be matched to the inner expansion of the outer solution.
Hydrodynamics of Slender Bodies
393
In the subcritical case, the outer solution is governed by Laplace’s equation (129) in transformed coordinates. The solution corresponding to
(134) near the body is a distribution of two-dimensional sources along the
body axis. This outer solution is identical to the thickness problem of twodimensional lifting-surface theory, and from (5.57) the appropriate source
strength is given by
q = −US ′ / h.
(136)
The desired source distribution, of strength (136), satisfying Laplace’s equation (129) in the transformed coordinates, or (126) in the physical coordinates (x, z), is of the form
φ=−
U
(1 − Fh2 )−1/2 ∫ S ′(ξ )log[(ξ − x )2 + z 2 (1 − Fh2 )]dξ.
4π h
l
(137)
Matching of this outer solution with (134–135) is assured by continuity,
and in view of (135) the arbitrary function f(x) in (132) can be obtained,
from (137), simply by setting z = 0. Thus, in the subcritical case Fh < 1,
f ( x) = −
1
(1 − Fh2 )−1/2 ∫ S ′(ξ )log ξ − x dξ.
2π h
l
(138)
In the supercritical case, the outer solution can be deduced from (131) by
noting that the velocity components are related linearly, in the form
∂φ ( 0 )
∂φ ( 0 )
= ±( Fh2 − 1)1/2
,
∂x
∂z
for z  0.
(139)
Equating this result to (134) in the overlap region, it follows that
∂φ ( 0 )
= −(US ′ / 2h )( Fh2 − 1)−1/2 ,
∂x
(140)
for |z| ≪ 1. Integrating both sides of this equation with respect to x, and
requiring that ϕ(0) → 0 as x → + ∞, the outer solution is obtained in the form
φ ( 0 ) = −(U / 2h )( Fh2 − 1)−1/2 S[ x + ( Fh2 − 1)1/2 z ].
(141)
Here the sectional area S(x) is defined to be zero, when the argument in
(141) is outside the range of the body length. Differentiating (141) with
respect to z confirms that this expression matches with the inner flux (134).
394
Chapter 7
Moreover, the arbitrary function f(x) of the inner solution (132) can be
found by setting z = 0 in (141). Thus, in the supercritical case Fh > 1,
f ( x) = −(1 / 2h )( Fh2 − 1)−1/2 S( x),
(142)
with the understanding that (142) vanishes if x is outside the range of the
body length.
With the function f(x) determined by (138) or (142), the inner solution
(132) is specified uniquely. The complete inner solution requires that the
boundary-value problem for Φ1 be solved. However, we shall show that
the leading-order hydrodynamic pressure on the body is dominated by the
function f(x). Thus the pressure forces acting on the body can be found to
leading order in ε, without solving for Φ1.
The hydrodynamic pressure can be obtained in the inner region from
Bernoulli’s equation (6.3). In the moving frame of reference, it follows that
∂φ 1
p = ρ U
− ∇φ ⋅ ∇φ  ρU 2[ f ′( x) − 21 ∇Φ1 ⋅ ∇Φ1 ].
 ∂x 2

(143)
Here the last approximation holds in the inner region, and f′(x) ≡ df/dx.
From (138) or (142) the function f(x) is of order S/h = O(ε), whereas the
boundary condition (133) gives the estimate ∇Φ1 = O( ε ). Thus the dominant contribution to the pressure on the body surface is due to the first
term in (143), or
p = ρU 2 f ′( x) + O( ε 2 ).
(144)
This approximation can be used to find the leading-order force and moment
acting on the body. Note from symmetry that the only force components
are the longitudinal drag force −Fx and the vertical force Fy. The only nonzero moment is the pitch component Mz.
The drag force due to wave resistance can be obtained by pressure integration from the formula
D = − Fx = − ∫∫ pnx dS = − ρU 2 ∫ f ′( x )S ′( x ) dx.
SB
(145)
l
Here we use (21) and the fact that the first-order pressure (144) is constant
across the body section.
In the subcritical case, the wave resistance (145) vanishes. This can
be confirmed directly, by substituting (138). The derivative of (138) is a
Hydrodynamics of Slender Bodies
395
Cauchy principal-value integral, and the result of substituting this in (145)
is a double integral that changes sign if the dummy variables are interchanged. Thus this double integral must vanish. More directly, since the
outer solution in the subcritical case is governed by Laplace’s equation,
with transformed coordinates, it follows from D’Alembert’s paradox that
there can be no drag force. In essence, there can be no wave resistance, to
leading order, since there are no waves in this outer solution.
In the supercritical case, the wave resistance is nonvanishing, essentially
because the outer solution (141) propagates to infinity without attenuation downstream. Formally, substituting (142) in (145), it follows that, for
Fh > 1,
D=
ρU 2
[ S ′( x)]2 dx.
2h( Fh2 − 1)1/2 ∫l
(146)
Note that this drag force is positive definite.
The vertical force and pitch moment can be obtained from pressure integration in the forms
Fy = ∫∫ pny dS ρU 2 ∫ B( x)f ′( x) dx,
(147)
M z ∫∫ pny x dS ρU 2 ∫ B( x )f ′( x )x dx,
(148)
SB
l
SB
l
where B(x) denotes the local waterplane beam.
In the subcritical case, it is convenient to integrate (147–148) by parts,
before substituting (138). Assuming that the beam vanishes at the bow and
stern, it follows that for Fh < 1,
Fy =
ρU 2
B′( x)∫ S ′(ξ )log ξ − x dξ dx,
2π h(1 − Fh2 )1/2 ∫l
l
Mz =
ρU 2
[ xB( x )]′ ∫ S ′(ξ )log ξ − x dξ dx.
2π h(1 − Fh2 )1/2 ∫l
l
(149)
(150)
In the supercritical case, (142) can be substituted directly in (147–148), and
thus for Fh > 1,
Fy = −
ρU 2
B( x)S ′( x) dx,
2h( Fh2 − 1)1/2 ∫l
(151)
396
Mz = −
Chapter 7
ρU 2
B( x)S ′( x)x dx.
2h( Fh2 − 1)1/2 ∫l
(152)
The “sinkage” and “trim” of a ship moving in shallow water can be computed by equating the vertical force and pitch moment to the hydrostatic
restoring force and moment (6.144) and (6.149). The resulting calculations
are of practical importance in predicting the “squat” of a ship, and ultimately the occurrence of grounding due to increased draft. A comparison of
these calculations with experiments is given by Tuck (1966) and reproduced
in figure 7.10. The theory is invalid near the critical Froude number Fh = 1,
but in other respects it agrees fairly well with the experiments. For typical
ships the vertical “sinkage” is the dominant effect in the subcritical regime,
whereas “trim” is dominant in the supercritical regime. These conclusions
follow immediately if one assumes fore-and-aft symmetry (see problem 15).
Figure 7.10
Sinkage and trim of a ship in shallow water vs. the Froude number (127). The solid
lines are experimental values, and the dashed lines are theoretical predictions obtained from (149–152). (From Tuck 1966)
Hydrodynamics of Slender Bodies
397
Various generalizations have been made of the original treatment by
Tuck (1966). For example, Beck, Newman, and Tuck (1975) treat the case of
a ship in a “dredged channel,” bounded on the sides by shallow water. An
interesting special case occurs when the flow in the channel is subcritical
while the flow in the shallow regions is supercritical. Other related problems are discussed by Beck (1977) and Tuck (1978), and in references cited
in these papers.
In the transcritical regime Fh ≃ 1, the linearized shallow-water approximation (125–126) is invalid. An appropriate nonlinear theory has been
developed by Lea and Feldman (1972), and generalized by Mei (1976) to
include dispersion as well as nonlinearities.
Problems
1. Derive from (26) the form of the outer potential at large distances from
a body in longitudinal motion. Compare this result with (4.123) and show
that the longitudinal added-mass coefficient is small compared to the displaced mass of the body.
2. Using the strip-theory results (40–42), compute the added mass and
added moment of inertia for a slender spheroid of length 2a and maximum
diameter 2b. Compare the results with figure 4.8, and find the values of a/b
where these approximations are in error by 10 percent of the exact value.
Does a spheroid satisfy the restrictions on the behavior of the sectional-area
curve at the two ends that have been assumed in deriving the slender-body
theory?
3. Using (35), derive an integral representation for the longitudinal added
mass of a slender spheroid. Show that as b/a → 0, the dominant term in this
expression is equal to
m11 =
4 b4 
a
πρ log + O(1) .
3
a 
b

Compare this approximation with the exact result shown in figure 4.8.
4. Following the approach outlined in problem 3, estimate the heave added
mass of the spar buoy shown in figure 2.16 with (1) the rigid free-surface
approximation and (2) the infinite-frequency approximation as illustrated
in figure 6.22. Compare these results with each other and with the observed
shift in resonance in figure 2.16.
398
Chapter 7
5. Show that the pitch and roll motions of a slender spar buoy are resonant
unless the centers of gravity and buoyancy coincide.
6. Explain the distinction between maneuverability and stability. In what
sense do these conflict?
7. Assume that a ship has elliptical sections and that the planform is a quadrilateral with draft T(x) = T0 − αx and vertical ends. Estimate the change in
the derivatives in table 7.2 due to a small trim angle α and the effect of
this trim angle on the stability parameter C. If xG = 0, CB = 0.7, T/L = 0.05
and B/T = 2.5, show that the ship will be stable if the bow-up trim angle is
greater than 0.7 degrees.
8. Using the same assumptions as in the derivation of equations (60–61)
derive the linearized equations of motion for a submarine in the vertical
plane. Include a static stabilizing moment kθ where θ is the pitch angle.
Show that the analogue of (72) for the stability roots is
Aσ 3 + Bσ 2 + [ C + k( m − ∂F2 / ∂U 2 )]σ − k( ∂F2 / ∂U 2 ).
Show that the submarine will be stable at all speeds if C > 0, and at low
speeds if C < 0.
9. Using the hydrodynamic derivatives for a streamlined body of revolution
with fore-and-aft symmetry without lifting surfaces, show that the critical
speed for a submarine to become unstable is
1/2
k( m + m22 )


U crit = 

m
m
m
m
−
+
(
)(
)
 22
11
11 
[ 2 g ( y B − yG )]1/2
where the approximation is valid if the body is slender.
10. Develop the linearized equations of motion for roll, pitch, surge, and
sway for an oceanographic research buoy falling with constant vertical
velocity U2, assuming that it is symmetrical about the x, y, and z axes,
except for the center of gravity which is at a position yG below the centroid.
What is the maximum terminal velocity for stability of the vessel in this
orientation, assuming m11 = m33 > m22?
11. Calculate the relative error incurred by the approximation (109) of the
Froude-Krylov force for a rectangular and triangular section in head seas of
various wavelengths. Confirm the statement following (109) regarding the
magnitude of this error.
Hydrodynamics of Slender Bodies
399
12. Rederive the strip theory results of section 7.7 for sway and yaw motions
of a ship, noting the similarities and differences with respect to heave and
pitch.
13. Show that the pitch and heave motions of a ship with fore-and-aft symmetry are uncoupled at zero forward speed. Is this true if U ≠ 0?
14. Show that the expression (37) holds for the sway force on a ship in shallow water, with a rigid free-surface condition, but not for the heave force.
15. Compute the sinkage and trim for a simple ship with parabolic waterlines, constant draft, and vertical sides by balancing the hydrodynamic
force and moment (149–152) with the hydrostatic restoring terms. Show
that the trim angle of the ship is zero for subcritical Froude numbers and
the sinkage (heave displacement) is zero for supercritical motion. Will a
ship operating at excessive speed in shallow water ground at the bow or at
the stern?
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Appendix: Units of Measurement and Physical Constants
A
Units
p
of
p
e
Measurement
n
and
d
Physical
i
x
Constants
© Massachusetts Institute of TechnologyAll Rights Reserved
The fundamental units of measurement in mechanics are mass (M), length
(L), and time (T). In the SI system these are measured in the basic units of
the kilogram (kg), meter (m), and second (s). Special names are given certain derived units. Thus the unit force of one Newton (N) is equal to one
kilogram-meter per second-squared, the unit of work or energy is one Joule
(J), equal to one Newton-meter, and the unit of power is one watt (W),
equal to one Joule per second. Prefixes denote decade factors such as the
kilo- (k) for 103, mega- (M) for 106, centi (c) for 10−2, and milli- (m) for 10−3.
Table A.1 lists various quantities in these and other units of measurement, and the relevant conversion factors. One nautical mile is the arc
length equal to one minute of arc on the surface of the earth; one knot is
the velocity unit equal to one nautical mile per hour. The (long) ton is equal
to 2240 pounds and the metric tonne is equal to 1000 kg.
Various physical constants are required in marine hydrodynamics, notably the density and kinematic viscosity of water, which are listed for various temperatures in table A.2 with the corresponding properties of air at a
pressure of one atmosphere. The standard acceleration of gravity (one g) is
equal to 9.80665 m/s2 or 32.174 ft/s2. The standard atmospheric pressure at
sea level is 1.01 × 105 N/m2 or 14.7 pounds per square inch. The surface tension of the interface between air and water varies between 0.076 N/m and
0.071 N/m for the temperature range 0–30°C, with the value 0.074 N/m
used commonly to describe capillary waves. The vapor pressure of water
increases from 610 N/m2 to 4230 N/m2 over the same temperature range, or
from 0.09 to 0.61 pounds per square inch.
Relative to these values for fresh water, the vapor pressure of salt water is
slightly reduced and the surface tension increased. Both changes are insignificant compared to the effects of temperature and impurities.
404
Appendix
Table A.1
Conversion Factors for Different Units of Measurement
Quantity
SI Unit
Other Unit
Inverse Factor
Length
1m
3.281 feet (ft)
0.3048m
1 km
0.540 nautical miles
1.852 km
Area
Volume
1m
2
1m
3
Mass
Force
0.0929 m2
35.315 ft
3
0.0283 m3
1 m3
264.2 gallon (US)
0.00379 m3
3
220.0 gallon (UK)
0.00455 m3
1 m/s
3.281 ft/s
0.305 m/s
1 m/s
1.944 knot
0.514 m/s
1 kg
2.205 pound
0.454 kg
1 Mg
0.984 ton (long)
1.016 Mg
1 Mg
1 tonne (metric)
1 Mg
lN
0.225 pound force
4.448 N
1 MN
100.4 ton force
9964 N
l MN
102.0 tonnef
1m
Velocity
10.764 ft
2
9807 N
0.000145 psi (pound per inch )
6895 N/m2
1J
0.738 foot pounds
1.356 J
lW
0.00134 horsepower
745.7 W
Pressure
1 N/m
Energy
Power
2
2
Table A.2
Density and Viscosity of Water and Air
Density, ρ, kg/m3
Temperature,
deg C
Fresh
Water
Kinematic Viscosity, ν, m2/s
Salt
Water
Dry
Air
Fresh
Water
Salt Water
Dry Air
0
999.8
1028.0
1.293
1.79 × 10−6
1.83 × 10−6
1.32 × 10−5
5
1000.0
1027.6
1.270
1.52
1.56
1.36
10
999.7
1026.9
1.247
1.31
1.35
1.41
15
999.1
1025.9
1.226
1.14
1.19
1.45
20
998.2
1024.7
1.205
1.00
1.05
1.50
25
997.0
1023.2
1.184
0.89
0.94
1.55
30
995.6
1021.7
1.165
0.80
0.85
1.60
Notes
N
N
o
o
t
t
e
e
s
s
© Massachusetts Institute of TechnologyAll Rights Reserved
Chapter 2: Model Testing
1. Advertisements for small toy racing cars typically claim a “speed of 760 scale
miles per hour.” One may question the basis for this scaling law, which certainly is
not that suggested by Osborne Reynolds!
2. It may not be obvious at this stage that the drag force on a body in an unbounded
fluid is independent of gravity. It will be shown in chapter 3 that the only role of
gravity is to introduce a hydrostatic pressure and a corresponding buoyancy force on
the body, which is additive to the hydrodynamic drag force.
Chapter 3: The Motion of a Viscous Fluid
1. Some authors prefer the symbol D/Dt, in equations such as (12) and (13) to
emphasize that (t ) is a material volume, but since these integrals are functions only
of time, we prefer the symbol d/dt in that context.
2. The roughness of the tube wall, ambient turbulence of the entering fluid, and
degree of smoothness of the transition into the tube are all important. Reynolds
observed the flow to remain at, or return to, the laminar state if the Reynolds
number, based on the tube diameter and mean velocity across the tube, was less
than about 2,000. With care, initially laminar fluid entering the tube would remain
laminar up to Reynolds numbers on the order of 12,000. Subsequent investigations
have attained much higher limits, as noted by Monin and Yaglom (1971).
3. This problem was first solved by Stokes, in 1851, as a means of estimating the
drag on a pendulum.
4. This assumption is valid because the governing equation is parabolic. Thus x < l is
a sufficient condition, excluding the complicated region near x = l. Physically, the
effects of viscosity are diffusive and the local solution in the boundary layer is independent of the flow downstream.
406
Notes
5. A rigorous justification for simultaneously changing both variables (U, x) to (uτ, δ)
requires that the Jacobian ∂(U, x)/∂(uτ, δ) be nonzero. This can be verified from the
empirical 1/7-th power relations to follow.
Chapter 4: The Motion of an Ideal Fluid
1. A strict interpretation of the material contour is essential to reconcile this statement with lifting-surface theory, and with other inviscid flows where vortices are
shed into the fluid at a sharp corner.
2. The interpretation of S + Sε as a single closed surface can be justified by connecting these two separate surfaces with a “tube” of infinitesimal radius, as shown by the
dashed lines in figure 4.6 (a); there is no contribution to the surface integrals from
this tube in the limit where the tube radius shrinks to zero.
3. Since the body is rigid and of constant volume, the source term of order r−1 must
vanish.
Chapter 5: Lifting Surfaces
1. Included here is a contribution proportional to (log A)/A2, which could not be
anticipated from Prandtl’s lifting-line equation (142).
2. This integral represents the downwash on the foil induced by the vortices in the
wake.
Chapter 6: Waves and Wave Effects
1. There is a semantic problem here involving the number of dimensions. We shall
refer to two- and three-dimensional motions of the fluid as in earlier chapters, corresponding respectively to one- and two-dimensionality of the free surface elevation
η(x, z, t).
2. A trochoid is the trajectory of any point on a circular disc rotating with its periphery in contact with a horizontal plane.
3. The derivative (114) and the resulting simplicity of (115) imply a fundamental
relation between the factor G″ in the stationary-phase approximation and the spatial gradient of the wavenumber. This relationship is displayed explicitly in an alternative derivation of the stationary-phase approximation given by Lighthill (1965).
4. It is clear that if the horizontal coordinates of the source position and observation point are reversed, the flow will be unchanged provided the direction of the
stream velocity U is also reversed. The dependence on the vertical coordinate and
source position is less elementary, but in the asymptotic approximation used here
both of these are in fact exponential.
Notes
407
5. The condition S13 = 0 is satisfied automatically for a body with a vertical plane of
symmetry, such as a ship, provided one of the horizontal coordinates lies in this
plane.
6. For finite depth h, these relations are still valid as ω → ∞, since for any fixed
depth the wavelength will be short compared to h. The symbol “o” in (175–176)
denotes that the order of magnitude of the exciting force is much smaller than that
of the quantity in parentheses.
7. Since nz is an odd function of z, the even part of ϕ7, which contributes to the
forces in the vertical plane, is a small first-order quantity.
8. Observed wave spectra, such as the Pierson-Moskowitz spectrum shown in figure
6.25, do not appear to satisfy this condition. Nevertheless the statistical distribution
of observed wave amplitudes tends to follow the Rayleigh distribution with sufficient accuracy to justify the conclusions that follow.
Chapter 7: Hydrodynamics of Slender Bodies
1. The weighted head of an arrow and the forward placement of automobile engines
are two common examples.
2. In the first-order solution, the appropriate radiation condition is that the waves
far from the body section should be outgoing. The situation is more complicated in
the higher-order solution, as discussed by Ogilvie and Tuck (1969).
3. This complication can be circumvented by introducing a mean effective depth
for each section, where the scattering pressure acts, and evaluating the exponential
factor for this particular depth, as outlined by, Gerritsma and Beukelman (1967).
4. Note that if (130) is extended to the subcritical case, one concludes that the
solution of (126) is a function of the transformed complex variable x′ + iz′ and its
conjugate.
Index
I
I
n
n
d
d
e
e
x
x
© Massachusetts Institute of TechnologyAll Rights Reserved
(Italicized page numbers denote reference to material in figures. Italicized numbers
in parentheses denote problem numbers.)
Abbott, I. H., 25, 26, 184
Abkowitz, M. A., 368
Abramowitz, M., 74, 237
Accelerating body, unsteady force on,
37–40, 38, 40
Acceleration, flat plate, in viscous fluid,
72–75, 75
Acosta, A. J., 218
Added mass
analogy with body mass, 147
of cylinder, 151–152, 153
definition, 147
Added-mass coefficients, 146
of circular cylinders, 151–152, 153
of ellipse, 152, 163(11)
of finned body, 163(10)
of flat plate, 152, 153, 162(5)
of floating body, 301
double-body, 309–310
high- and low-frequency limits,
309–310
of particular body forms, 310, 311,
312
symmetry, 308
flow past, 161
general properties, 147–154
relation to dipole moment, 149–151
relation to exciting force, 315
relation to kinetic energy, 148
of simple forms, 151–154, 152
of slender body, 153
of sphere, 151
of spheroid, 152, 153, 154, 163(8)
of square, 152, 153, 163(10)
symmetry, 148, 153
of two-dimensional bodies, 144–154,
153, 162 (7), 163(10), 163(11),
163(12)
Added-mass tensor, 146
Added moment of inertia
of circle, 152, 152
of flat plate, 152, 152, 163(9)
spheroid, 153, 154, 163(9)
of square, 152
Adee, B. H., 47
Advance ratio, 26, 28
Air, density and viscosity coefficients,
3–4
Aircraft carriers, motion in waves,
48–49, 49, 374
Air-water interface, 14
Alternating tensor, 145
Amplitude dispersion, 260
Analytic function, 123–129, 124, 162(6)
Angle of attack
flat plate, 243(6)
410
Angle of attack (cont.)
of hydrofoils, 50(4)
induced, 212–213
of propellers, 26, 27, 28
relation to lift coefficient, 180–182
of unsteady hydrofoil, 244(16)
Apparent mass and damping, 43
Archimedes’ principle, 9–11, 305
Argand diagram, 237, 238
Aspect ratio, 167–168, 214
Aspect-ratio lifting surface, 167
Asymptotic approximation, 210,
212–216
Atmospheric pressure, 66
ATTC (American Towing Tank
Conference) friction coefficient,
33
Axisymmetric body
flow past, 88, 117–120, 118, 121–123,
122
heave response, 46, 47
Bark, F. H., 64
Beach, waves incident upon, 50(1)
Beam-length ratio, 153
Beam waves, 377
Beaufort scale of sea state, 330
Beck, R. F., 397
Berger, E., 41
Bernoulli’s equation, 3, 112–114, 119,
139–141, 174, 249, 301
for inner region hydrodynamic
pressure, 394
for lift force, 227, 234
linearized, 177, 218, 335(2), 378
wave energy, 274
Berteaux, H. O., 48
Bessel functions, 237, 240
Biot-Savart law, 200
Birkhoff, G., 10
Bishop, R. E. D., 50, 323, 326, 327, 330,
334, 368
Bisplinghoff, R. L., 238
Index
Blasius, H., 78–79
Blasius laminar boundary layer, 19, 79,
81, 88, 103(5), 103(6)
Body force, 66–67
Body-mass force, 147, 155–156
on nonuniform stream, 156–159
on spar buoy, 370–372, 371
of spheroid, 164(14), 164(15)
Body motions
comparison with mechanical
oscillator, 322
effect of mooring on, 48
of fixed vs. moving bodies, 132–133,
133
in horizontal plane, 333
in irregular waves, 297, 333
in regular waves, 46, 319–323
of ships in waves, 45–50, 369–388
in unbounded fluid, 141–147
unsteady, 133
vertical heave oscillations, 45–48
Body potential, 369–371
Body of revolution. See Axisymmetric
body
Boundary condition
on body in waves, 384
body-wall interactions, 160–161
dynamic, 66, 114, 115–116, 248,
249–250
on fixed body, 132–133, 133
flat plate, 162(6)
in flow between parallel walls, 67–68
free surface, 66, 116, 248–250, 254
on hydrofoil, 172–176
in ideal fluid, 114–116
kinematic, 66, 67, 114–115, 248–249,
250
on lifting surface, 178–180
linearized, 222, 224–225
on moving body, 143–144
relation to complex potential, 161(1)
in shallow water, 392
on slender body, 343–344, 345
Index
of unsteady hydrofoil, 231, 239
in viscous fluid, 72, 73
Boundary layer, 34–35
approximation, 70–71
of ideal fluid, 107
laminar, 75–88
steady two-dimensional flow on,
81–88
three-dimensional, 88
thickness, 74–76, 80–81
turbulent, 17, 81, 91–102, 103(6)
in unsteady force, 38–39
in viscous flow, 7, 55–56
Boundary-layer theory, 6–7
Bow, wave resistance, 296
Bow waves, 377
Brard, R., 200
Brennen, C., 222, 239
Brokaw, C. J., 239
Buoyancy
center of, 303–306
neutral, 306
Buoyancy force, 15, 43, 159, 303
Burkart, M. P., 228
Cables, mooring, 48, 52(12), 151
Camber, 169, 175–176, 212–213
Canoes, propulsive equilibrium, 52(11)
Cape Cod Canal, 103(3), 103(4)
Capillary waves, 247
Carrier, G. F., 189
Carrier waves, 268, 269
Cartesian coordinates, 56, 57, 64, 248–
249, 289
Cascade effect, of propeller, 168
Cauchy-Poisson problem, 267–268
Cauchy principal-value integral, 179,
394–395
Cauchy-Riemann equations, 124, 226
Cavitation, 4–5
hydrofoils, 51(9)
number, 4–5, 216–217
propellers, 28
411
Cavity flow, 216–218
symmetric, 218–223, 223, 224
Cebeci, T., 102
Center of buoyancy, 303–306
Center of flotation, 304–305
Center of gravity, 305–306
in dynamic stability, 367
Center of pressure, 180
Chain rule, 62, 74, 109
Chord length, 23
Circle, in ideal flow, 133, 133
Circular cylinder
added-mass coefficients, 151–152, 153
flow, 160
force coefficient, 40, 40
in ideal flow, 121
moving between walls, 160–161
splitter plates, 41
Strouhal number, 40–41
unsteady flow past, 38, 38–40, 40
vortex shedding, 40–41
Circulation
about hydrofoil, 169–171, 170, 171
conservation along vortex filament,
198, 198
conservation in ideal fluid, 108–110
Complex potential, 152
for added-mass coefficients of flat
plate, 162(5)
in conformal mapping, 125–128, 126
in ideal fluid motion, 123–125, 124,
125
relation to boundary condition, 161(1)
of source, 186
of streamline, 123–125, 124, 125
of velocity field, 123–125, 124, 125
velocity potential, 123–125, 124, 125
of vortex, 186, 243(12)
Complex variables, 186–189
Conformal mapping, 125–129, 126
complex potentials in, 125–128, 126
of supercavitating hydrofoil, 225,
225–226
412
Conservation
of mass, 55, 58–59, 61
of momentum, 55, 58–59, 62, 65, 67,
83–84
Continuity equation, 6, 61–62, 65,
68–69, 89–90, 107, 111–112, 121
Coriolis acceleration, 247
Corner flow, 124, 124–126
Correlation allowance, 34
Couette flow, 67–68, 69, 71, 83, 102(1)
Cox, G. G., 168
Cramer’s rule, 364
Cross-coupling coefficients
added mass, 147
heave and pitch, 382–383
Cross-coupling effects, 321
Cummins, W. E., 141, 323
Cylindrical coordinate systems, 129–132
Dagan, G., 294
D’Alembert’s paradox, 142, 354, 395
Damping, 306–307
viscous, 372, 373
Damping coefficient, 301, 308–312
high- and low-frequency limits, 309,
309, 310, 317
of particular body forms
rectangular cylinder, 311
sphere, 312
relation to energy flux, 308
relation to exciting force, 315
symmetry, 308
Davis, A. M. J., 317, 321
Delta wing, 356
Density
of air and water, 3–4
of viscous fluid, 61
of wave energy, 325–326
Design extreme value, 332
Diffraction potential, 314, 315–317. See
also Scattering potential
Diffraction problem, 299
Diffusion equation, 72
Index
Dimensional analysis, 7, 9–15
of lifting surfaces, 22, 22–24, 25
of propellers, 25–28, 29
of unsteady force, 37
of waves, 12–14
Dipole, 120, 124, 131, 132, 136
moment, relation to added-mass
coefficients, 149–151, 163(11)
Dirac delta function, 178
Directional stability, 365–367
Dispersion, 323–324
Dispersion relation, 252
finite-depth, 254
group velocity, 269
linear, 260
nonlinear, 259–260
Displaced mass coefficients, of
two-dimensional bodies, 162 (7)
Displacement thickness, 79–81, 82,
83–84, 85, 103(5)
Divergence theorem, 59, 115–116, 140,
148, 207
Dividing streamline, 118, 118, 119
Doenhoff, A. E. von, 25, 26, 184
Doppler shift, 323, 374, 377
Double-body approximation, 309–310
for ship maneuvering, 358
Doublet, 120
Downwash, 230, 236
effect on lift, 206
Drag
cross-flow, 48, 360
effect of separation on, 21
effect of turbulence on, 20–21, 21
frictional, 18, 19, 20, 23–24
on general bodies, 18, 20–22
on hull, 29–34, 32, 33
hydrofoil, 184–186
on hydrofoils, 22, 22–24, 25
induced, 206–210
pressure (form), 20, 23–24
separation of, 15, 16, 17, 21
on sphere, 14–17, 16
Index
steady-state, 15
on streamlined bodies, 18, 20–21
on submarines, 50(3)
total, 18
unsteady hydrofoil, 239
viscous, 17–21, 19, 21
Drag coefficient, 15, 16, 17
of hull, 30
hydrofoils, 23–24, 26
relation to Froude numbers, 32, 33
of sphere, 102
steady-state, 37, 39
for three-dimensional cylinders, 20
total, 20
for two-dimensional cylinders, 20–21,
21
Dredged channel, 397
Dynamic boundary condition, 66
Dynamic pressure, 67, 114, 138–141
Dynamic similitude, 5, 7
Eatock Taylor, R., 368
Effective mass, 39
Effective volume, 159
Efficiency
hull, 36
propeller, 26, 28, 51(7)
propulsive, 36
relative rotative, 36
Eggers, K., 296
Ellipse, added-mass coefficients,
151–152, 152, 153, 163(11)
Ellipsoid
added-mass coefficients, 310, 312
damping coefficient, 310, 312
flow past, 132
Elliptic planform, lift coefficient,
214–215, 215
Energy, of waves, 271–277, 335(4)
density, 325–326
flux, 274–276, 275, 278
relation to damping coefficient, 308
ship waves, 290–292, 296
413
spectrum, 326
directional, 326–327
JONSWAP, 331
Pierson-Moskowitz, 328, 329, 330,
330–331
Equations of motion
of body in waves, 320–323
ship maneuvering, 363–368
Error function, 74
Euler’s equations, 62, 65, 90, 107,
112–113, 159
Exciting force, 313–319
Froude-Krylov hypothesis, 317–319,
370–371, 372, 373, 383–384,
386
long-wave approximation, 313–314
relation to added-mass coefficient,
315
relation to damping coefficient,
315
relation to Haskind relations, 386–387
strip-theory approximation, 381,
383–384, 386
of thin-ship, 317–318
External flow, 132
Extrapolation, 31, 33, 33
Extreme-value statistics, 331–332
Faltinsen, O., 382, 383, 386, 387
Far-field fluid motion, 141, 149–151
Favored pressure gradient, 82–83, 83
Feldman, J. P., 397
Feshbach, H., 110, 129, 132
Fetch, 327–328, 331
Field point, 134, 135
Fink, P. T., 357
Finned body
added-mass coefficient of, 163(10)
added moment of inertia, 152, 152,
163(9)
First-order approximation, 173
Fish-line problem, 277
Flapping propulsion, 239
414
Flat plate
acceleration in viscous fluid, 72–75, 75
added-mass coefficients, 152, 153,
161, 162(5)
added moment of inertia, 152, 152,
163(9)
angle of attack, 243(6)
boundary condition, 162(6)
drag force, 186
flow past, 128
frictional-drag coefficient, 30
friction resistance, 6
heave, 239
hydrofoil, 23–24, 180–184, 181, 186
ideal flow past, 124–125, 125
laminar boundary layer, 75–88
lift coefficient, 181, 243(5)
steady-state flow past, 75–81
turbulent boundary layer, 81, 91–99
roughness effects on, 100–101, 101
unsteady motion of, 72–75
viscous drag, 17–18, 20, 21
viscous flow past, 70–71
between walls, 161
FLIP, motion in waves, 333, 334
Fluid dynamics, 2
Fluid motion, balance of forces in, 3–4
Fluid viscosity, 14
Following waves, 317, 377
Force. See also Body-mass force
on body in nonuniform stream,
156–159
hydrodynamic pressure, 67, 114,
138–141
orders of magnitude, 2–3
Force coefficient, of circular cylinders,
40, 40
Form drag
relation to residual-drag coefficient, 33
relation to Reynolds number, 31, 31,
33–34
Fourier coefficient, 209
Fourier series, 196, 213
Index
Fourier-Stieltjes representation, 325
Fourier theory, 267
Frank, T., 368
Free-surface boundary condition
linearized, 248–250
method of images approach, 147
nonlinear, 256–258
wave motions on, 67
Free-surface elevation, 250
Free-surface flow, 116
Free-wave distribution, 283
Frequency of waves, 247
deep-to-shallow, 256, 257
relation to frequency of encounter,
386
relation to wavenumber, 251–252
Frictional drag, 18, 19, 20, 23–24, 64
Frictional-drag coefficient, 18, 19, 20,
34, 81, 98
local, 98
in self-propelled test, 34
Frictional resistance, 6
Friction velocity, 92
Froude hypothesis, 6, 33, 294, 295
experimental validation, 30–31, 32
Froude-Krylov exciting force, 317–319,
370–372, 371, 373, 383–384, 386
Froude number, 4, 5, 6, 7, 48–49, 280,
281
for hulls, 30
relation to drag coefficient, 32, 33
relation to scale, 30
for shallow waves, 389–390
subcritical, 290
Fully developed wave spectra, 327–328
Fully wetted flow, 190, 226
Gadd, G., 294
Gauss’s theorem, 139–140, 157, 303,
314
General force equation, 206
Geosim, 18, 32, 33
Gerstner, F. J. von, 260–261
Index
Glaubert integral, 195, 209
Goldstein, S., 16
Granville, P. S., 99
Gravitational acceleration, 5, 11, 66–67
Gravitational force, orders of
magnitude, 2–3
Green function, 137–138
Green’s theorem, 127–138, 141–142,
188, 308
applied to dipole moments, 149–150
and Haskind relations, 315–316
for representation of velocity
potential, 162(4), 163(13)
surfaces of integration, 135
Griffin, O. M., 41
Group velocity, 268–271, 269, 270, 272,
283
relation to wave energy flux, 276–277
Gust problem, 239–240, 240, 241, 242
Haberman, W. L., 28
Half body, 117–118, 118, 162(3)
Halfman, R. L., 238
Hankel function, 237
Haskind, M. D., 323
Haskind relations, 315–316, 319, 372,
373, 386–387
Häusler, F. U., 48
Havelock, T. H., 132, 290, 294
Head waves, 377
Heat equation, 72
Heave. See also Body motions
added-mass coefficient, 312
damping coefficient, 312, 318
definition sketch, 298
effect on axisymmetric bodies, 46, 47
equations of motion, 373–374
exciting force, 318
high-frequency limit, 310
low-frequency limit, 309
oscillatory rigid-body motion, 238
strip theory, 375, 378–387
vertical oscillations, 45–48
415
Heave-amplitude ratio, 46, 47, 48
Helmholtz’ theorem, 110
High-speed ships, wave resistance, 296
Hilbert transform, 194–195, 221
Hildebrand, F. B., 121, 187
Hinch, E. J., 64
Hogben, N., 44
Horseshoe vortex, 204, 205, 205–206
Houmb, O. G., 331
Huang, T. T., 102
Hughes friction coefficient, 31, 33
Hughes line, 31, 33, 33
Hull
double-body image, 358
drag, 29–34, 32, 33, 292–296
efficiency, 36
frictional resistance, 6
wetted-surface area, 29
Hydraulic smoothness, 100
Hydrodynamic pressure, 67
Hydrofoil
angle of attack, 50(4)
cavitation, 51(9)
chord length l, 23
dimensional analysis, 22, 22–24, 25
drag, 22, 22–24, 25
large aspect-ratio, 210–216
lifting problem, 176, 176–184
NACA a series, 184, 184, 185
“nose-tail line,” 22
planform area, 23, 24
scaling laws, 51(10)
span, 23
steady-state, 171–172
supercavitating, 217, 217–218,
223–236
thickness problem, 138, 175–176, 176,
186–189
three-dimensional, 24
two-dimensional, 23, 24, 25, 168–172
drag force, 184–186
linearized, 172–176
vortex distribution, 186–189
416
Hydrofoil (cont.)
unsteady, 230–236
gust problems, 239–240, 240, 241,
241
oscillatory time dependence,
236–239, 241
transient problems, 241–242, 242
Hydrostatic force, 302–312
relation to exciting force, 314–315
total, 305
Hydrostatic pressure, 67, 103(3), 114
Ideal fluid, 6, 38–39
boundary condition, 114–116
boundary layer, 107
Ideal fluid motion, 107–165
conformal mapping of, 125–129,
126
Green’s theorem of, 127–138
hydrodynamic pressure forces,
138–141
irrotational flow, 108–110, 146–147
simple potential flow, 116–120
stream function, 121–123, 122
in unbounded fluid, 141–147
Images, method of, 147, 160–161,
243(9), 309–310
Impulse-response function, 323
Incident waves, 297–301, 371–373
pressure field, 378
velocity potential, 298
Inertial force, 319, 320
balance with viscous gravitational
forces, 5
relation to viscous force, 42, 44
Inertial waves, 247
Inner law, of turbulent boundary layer,
93
Inner solution
high aspect-ratio lifting surface, 211
horizontal force on spar buoy,
372–373
ship in shallow water, 391–392, 394
Index
slender body, 347, 369–374
turbulent boundary layer, 91–92
Integral equation, 179, 186
singular, 189–197, 233–234
Integro-differential equation, 213
Interaction force, 160, 164(15)
Interference of waves
from bow and stern, 280
in convoy, 281
three-dimensional, 280–281, 290–292
Internal flow, 163(13)
Internal waves, definition, 247
Inui, T., 294
Inviscid flow, 184–185
Inviscid fluid, 6–7
Inviscid limit, 37
Irrotational flow, 108–110, 146–147
steady-state, 112–113
unsteady, 113–114
velocity potential, 110–112
ITTC (International Towing Tank
Conference) friction line, 31, 33,
33, 51(8)
JONSWAP (Joint North Sea Wave
Project), 331
Joubert, P. N., 99
Kajitani, H., 294
Kármán, T. von, 40, 84, 90
Kármán street, 40–41
Keel, lift and drag forces, 243(11)
Kelvin, Lord, 159, 283, 285–286
Kelvin’s method of stationary phase,
277
Kelvin’s ship-wave pattern, 283, 285,
286, 286–287, 293
Kelvin’s theorem, 108–110, 112, 170,
171, 229, 232
Kennard, E. H., 152
Kerczek, C. H. von, 102
Kerwin, J. E., 168
Kibel, I. A., 153
Index
Kim, W. D., 311–312
Kinematic boundary condition, 66, 67
Kinematic viscosity coefficient, 14
Kinetic energy, relation to added-mass
coefficients, 148
Kinoshita, M., 290
Kinsman, B., 327
Kochin, N. E., 153
Konig, J. G., 28
Korvin-Kroukovsky, B. V., 50, 341
Kostyukov, A. A., 294
Kramers-Kronig relations, 312
Kroenecker delta function, 150, 156,
305, 320
Krook, M., 189
Kussner function, 241, 242
Kutta condition, 172, 176, 179, 184,
185, 191, 193, 196, 201, 220, 226,
233
Kutta-Joukowski theorem, 175, 177,
242(3)
Kux, J., 102
Lagally theorem, 141
Lagrangian coordinates, 262
Laitone, E. V., 260, 261, 268, 277, 293
Lamb, H., 132, 136, 159, 161, 261, 268,
277, 289
Laminar flow, 18
definition, 70
between parallel walls, 67–68, 69
past flat plate, 70–81
in pipe, 68–70
on sphere, 16, 17
Lammeren, W. P. A. von, 28
Landahl, M. T., 64
Landau, L. D., 88
Landweber, L., 141
Laplace equation, 112, 115–117,
129–130, 132, 134, 137, 157,
233–234, 250–251, 258, 300, 345,
384
wave energy, 274
417
Laplace transform technique, 74, 241
Laurent series, 131, 215
Laurmann, J. A., 210
Law of Biot-Savart, 200
Law of the wall, 93
Lea, G. K., 397
Leading-edge suction force, 186
Legendre functions, 130–131
Le Méhauté, B., 255
Length (L), 10
Levi-Civita, T., 260
Lewis, E. V., 50, 327
Lifshitz, E. M., 88
Lift, 22
Lift coefficient, 356
due to angle of attack, 23–24, 180–181
elliptic planform, 214–215, 215
flat plate, 243(5)
hydrofoil, 23, 24
supercavitating flow, 223–236
two-dimensional, 243(7)
of uncambered foil, 181
Lift-drag ratio, 208–210
Lift force
horizontal, 22
oscillatory, 41
sectional, 24, 25, 26
slender body, 354–357
for unsteady hydrofoil, 234–235
in vortex shedding, 40–41
Lifting-line equation, 213
Lifting-line theory, 210–216
for slender body, 356
Lifting problem, 189–197
hydrofoils, 176, 176–184
relation to vortex distribution,
177–180
three-dimensional planar surface,
200–206
unsteady hydrofoil, 230–236
Lifting surfaces, 1–2, 167–245
chord length, 167
definition, 167
418
Lifting surfaces (cont.)
dimensional analysis, 22, 22–24, 25
span, 167
three-dimensional, 168, 168
Lighthill, M. J., 260, 353
Linearization
of Bernoulli equation, 218
of body motion in waves, 46, 48, 300
of body motions in water, 43
of cavity flow, 218, 219
of free-surface boundary condition,
248–250
of paddlewheels, 103(8)
thin-wing theory, 159
of wavelength, 13
of waves, 13
Long-crested waves, 327
Longuet-Higgins, M. S., 263
Long-wave approximation, 313–314
Low-aspect ratio lifting-surface theory,
356
Low-Froude-number asymptomatic
tendency, 31, 32
Lucy Ashton drag coefficient, 32
Lumley, J. L., 64
Mach cone, 290
Mach number, 4, 390
Mair, W. A., 41
Mandel, P., 358, 360, 361
Manen, J. D., 28
Mangler, W., 88
Mariner class ship, motion in waves,
387, 388
Maskell, S. F., 323
Mass, conservation of, 55, 58–59, 61
Mass transport, 261–263, 335(3)
Matched asymptotic expansions,
94–95
in lifting-line theory, 214
in slender-body theory, 347, 348–351
Material volume, 58
Matheson, N., 99
Index
Mean-camber line, 175–177, 178, 180,
182, 184, 185, 231
oscillatory, 238
Mean velocity profile, 95–96, 96
Mechanical oscillator, comparison with
floating body, 322
Mei, C. C., 263, 397
Memory effect
on floating body, 322
on unsteady hydrofoil, 230–231, 235,
237
Metacentric height, 306
Michell’s integral, 293–294, 295, 319
Michell theory, 138
Milgram, J. H., 44
Milne-Thomson, L. M., 128, 141
Michell, J. H., 292
Michell’s integral, 293–294, 295
Moment. See also Force
on body in waves, 313–315, 316–317
hydrofoil, 175
of wave spectrum, 328, 330
Momentum, conservation of, 55, 58–59,
62, 65, 67, 83–84
Momentum thickness, 80, 85, 86–87,
97–98, 99–100
Monin, A. S., 95–96, 96, 326
Moore, C. S., 306
Moored-body motions, 48, 52(12), 151,
297
Morgan, W. B., 168
Morison’s formula, 44, 51(14)
Morse, P. M., 110, 132
Motion of bodies in fluid. See Body
motions
Motora, S., 361, 362
Multipoles, 132, 149
Munk moment, 355, 367
Muskhelishvili, N. I., 189, 221
NACA a series hydrofoils, 184, 184, 185
Navier-Stokes equations, 6, 65–69,
75–76, 107
Index
applied to turbulent conditions,
88–89
Near-wall approximation, 96, 97
Neumann, G., 247, 326, 327, 330
Neutral buoyancy, 306
Newman, J. N., 152, 161, 287, 316, 342,
357, 373, 375, 376, 386–387, 391,
397,
Newtonian fluid
conservation of momentum in, 65
stress relations in, 62–64, 63
Newtonian mechanics, 10–11
Newton’s equation, 5–6, 147, 155
Noblesse, F., 294
Nonuniform stream, 156–159
Norrbin, N. H., 368
“Nose-tail line,” of hydrofoils, 22
Ocean waves. See Random waves
Ochi, M. K., 332
Octapoles, 132
Ogilvie, T. F., 312, 342, 382–383
1/7-power approximation, 99–100,
103(6)
Oosterveld, M. W. C., 28
Open-water characteristics, of
propellers, 35, 36
Open-water tests, of propeller-hull
interactions, 35, 36
Outer solution
high aspect-ratio lifting surface,
211
ship in shallow water, 391, 392–393
slender body, 347
Overik, T., 331
Overlap region, 94–95
Paddlewheels, 103(8)
Parabolic mean-camber hydrofoil,
182–184
Parabolic velocity profile, 68
Parkinson, A. G., 368
Pearson, C. E., 189
419
Pendulum, 11–13
Pendulum angle, 13
Phase velocity
finite-depth, 254–255, 255
linear, 252–253
nonlinear, 259–260
relation to wavelength, 13–14
shallow water, 255–256, 257
of waves, 252–253, 253, 255, 255
Phillips, O. M., 247, 261, 326
Pierson, W. J., 247, 326, 327, 330
Pierson-Moskowitz spectrum, 328, 329,
330, 330–331
Pilings
added-mass coefficients, 151
wave force on, 44
Pipe flow, 68–70
Pitch. See also Body motions
definition sketch, 298
equations of motion, 373–374
high-frequency limit, 310
low-frequency limit, 309, 309
oscillatory rigid-body motion,
238
of propellers, 28, 29, 51(7)
of ship, 48–49, 49
strip theory, 375, 376, 378–387
static stability, 306
Pi theorem, 10
Planar lifting surface, 23, 138, 168,
341–342
three-dimensional, 200–206
Planform area, 23, 24
Pohlhausen, K., 85, 87
Pohlhausen velocity profile, 87
Poiseuille flow, 68–70
Prandtl, L., 6–7, 38, 38, 70, 95, 210, 213,
214, 216
Prandtl-Glaubert transformation, 390
Pressure
dynamic, 67, 114, 138–141
force, 155–156, 157, 159
hydrofoil, 174–175
420
Pressure (cont.)
gradient
effect on boundary layer, 82–83, 83,
85
of laminar boundary layer, 85–86
relation to buoyancy force, 43
semiempirical analysis, 102
stress, 20, 62, 64
Price, W. G., 50, 323, 326, 327, 330, 334
Principal-value integral, 187–188
Probability density function, 328
Progressive waves, 250–253
deep-water, 253, 253, 256, 258–260
electric power from, 335(5)
energy, 271–277, 272, 273
finite depth, 253–256, 335(3)
mass transport, 261–263
nonlinear, 256–261
periodic, 12, 12
regular, 297–301
Propeller, 25–28, 27, 29, 50(5), 50(6)
advance ratio, 50(6)
angle of attack, 26, 27, 28
cavitation, 28
dimensional analysis, 25–28, 27, 29
efficiency, 26, 28, 51(7)
flow past, 168
open-water characteristics, 35, 36
pitch, 51(7)
self-propelled test of, 34–36, 36
shaft, 25, 27
supercavitating, 28, 36
Propeller-hull interactions, 34–36
open-water tests, 35, 36
Propulsion, flapping, 230
Propulsive efficiency, 36
Propulsive equilibrium, of canoes,
52(11)
Pseudo-velocity function, 190–191
Quadrupoles, 132, 149
Quarter-chord point, 181, 228, 235–236
Quartering waves, 377
Index
Radiation condition, 300–301
Radiation problem, 299
Ramberg, S. E., 41
Random waves, 297, 323–332
Rankine ovoid, 118–119, 119, 162(2)
Rawson, K. J., 306
Rayleigh distribution, 328, 331, 333
Rectangular planform lifting surface,
356
Relative motion hypothesis, 315
Relative rotative efficiency, 35
Residual-drag coefficient, 30
relation to form drag, 33
Residual resistance, 6
Response amplitude operator, 321
Reynolds number, 3, 4, 5, 6, 15, 16,
17–18, 20, 21, 70
for boundary-layer thickness, 80
full-scale, 51(8)
hydrofoil, 23–24
of laminar flow, 70
local, 81, 98
relation to form drag, 31, 31
relation to Strouhal number, 41
of unsteady force, 37
wave force effects on, 42
Reynolds stress, 90–91, 93–95
Riemann mapping theorem, 128–129
Riemann sheet, 226
Right-hand rule, 201, 208
Rigid body, 141–147
Rigid-body motion, oscillatory, 238
Rigid-wall boundary condition,
309
Ripples, 247
Robinson, A., 210
Roll
definition sketch, 298
high-frequency limit, 310
low-frequency limit, 309, 309
of ship, 48–49, 49
static stability, 306
Roshko, A., 41
Index
Roughness
allowance, 34, 101
effect on turbulent boundary layers,
100–101, 101
Roze, N. V., 153
Rudder, slender-body theory of motion,
361–362
Rudder angle, 358, 363, 365
Rudder force and moment, 361–363
Rudnick, P., 333
Salt water, effect on boundary-layer
thickness, 103(7)
Salvesen, N., 382, 386, 387, 388
Sarpkaya, T., 40, 40
Scalar, 122
Scale models, for full-scale phenomenon
prediction, 3
Scattering effect, 299
Scattering potential, strip-theory
approximation, 383–384, 386
Schlichting, H., 81, 88, 95–96
Schoenherr formula, 99
Schoenherr frictional-drag coefficient,
18, 19
Schoenherr line, 31, 32, 33
Schwartz, L. W., 260
Schwarz-Christoffel transformation, 128
Schwarz reflection principle, 226
Screw propeller. See Propeller
Seakeeping. See Body motions; Ship
motions in waves
Sears function, 240, 240
Sedov, L. I., 10, 161
Self-propelled test, 34–36, 36
Separation
of circular disc, 21
effect of turbulence on, 17
effect on drag, 15, 16, 17, 21
of flow, 56
around hydrofoil, 169–170, 170
pressure gradient-related, 82–83, 83
relation to stall, 24, 25
421
of variables, 129–132
viscous, 358
Shallow-water, effect on ships, 388–397
Shallow-water waves, 255–256, 257, 389
Shape factor Λ, 86, 97
Sharma, S., 296
Shear flow, 63
Shear stress, 86
viscous, 46
on wall, 97
Ship maneuvering, 357–388
equations of motion, 363–368
experimental comparison, 360–361,
361
force coefficients, 360
notation, 360
rudder force, 361–362
slender-body theory, 358–362, 360
steady turn, 363–365, 364
Ship motions in waves, 369–388
strip theory, 374–397
Ship waves
drag
relation to energy radiation, 290–292
thin-ship theory, 292–294
wave breaking-related, 296
energy flux, 290–292
stationary phase, 285–290, 292
three-dimensional, 282–286, 290–292
two-dimensional, 277–281
Shoaling coefficient, 276
Similarity parameter, 74
Similarity solution, of boundary-layer
equations, 78–79
Singularity, 116–120. See also Sink;
Source
distributions, 133–138
two-dimensional, 120
Sink, 116–118
differentiated from source, 117
relation to stream function, 123
Sinkage and trim, in shallow water, 396,
396
422
Slender-body theory, 341–401
added-mass coefficients, 153
for body in infinite fluid, 369
boundary conditions, 343–344, 345
lateral flow, 342–344, 346, 347,
351–357
lateral force, 346, 351–357
longitudinal flow, 344, 345–346
low-aspect-ratio lifting-surface theory,
356, 359
of rudder motion, 361–362
in shallow water, 388–397
for ship maneuvering, 358–362,
360
unbounded ideal fluid, 342–348
Slenderness parameter, 342, 379–380
Smith, A. M. O., 102
Smith, W. E., 388
Smoke ring, 197
Society of Naval Architects of Japan,
294, 296
Söding, H., 48
Soh, W. K., 357
Source
differentiated from sink, 117
distribution, 188
potential, 116–120
related to Green function, 137–138
relation to stagnation point, 117–118,
118
relation to stream function, 123
strength, 117, 134
on symmetric strut, 220
Source distribution, as defined by
Green’s theorem, 127–138
Source point, 134
Source-sink combinations. See Rankine
ovoid
Span, hydrofoil, 23
Spar buoy, motions in waves, 46, 47,
53(15), 102(2), 333–334, 370, 371,
371–373
Spectral energy density, 326
Index
Sphere
added-mass coefficients, 151, 312
damping coefficients, 312
drag, 14–17, 16
drag coefficient, 102
ideal flow on, 129–132, 133, 133
laminar flow on, 16, 17
moving near wall, 160
Spheroid
added-mass coefficients, 152, 153,
154, 163(8)
added moment of inertia, 153, 154,
163(9)
body-mass force, 164(14), 164(15)
Splitter plates, 41
Square
added-mass coefficient of, 163(10)
added-mass coefficients, 152, 153
added moment of inertia of, 152
Squat, 396
Stability
directional, of ship, 365–367
excessive, 367
indices, 365–366, 366
of submarines, 368
of supertankers, 368
Stagnation, relation to source, 117–118,
118
Stagnation point, 15, 17, 117–118
relation to stream function, 123
Stall, 181
relation to separation, 24, 25
Standing waves, 335(2), 335(4)
Starting vortex, 203, 230, 233, 235
Stationary bodies, wave force on, 41–44
Stationary phase approximation, 285,
286–290, 316
Stationary waves, 325
Steady-state drag coefficient, 37, 39
Steady-state flow, past a flat plate, 75–81
Steady translation problems, 142–143
Stegun, I. A., 74, 237
Stoker, J. J., 268, 289, 301
Index
Stokes’ drift, 262
Stokes’ equations, 260
Stokes’ theorem, 109–110
Stokes’ wave profile, 261
Storms, waves generated by, 325, 327,
330
Stream function, 121–123, 122
Streaming flow. See Streamline
Streamline, 113, 162(3)
complex potentials, 123–125, 124, 125
past semi-infinite half-body, 117–118,
118
Rankine ovoid-associated, 118–119,
119
two-dimensional hydrofoil, 169,
169–170, 170
Streamlined body
drag on, 18, 20–21
flow past, 82
Stress relations
Newtonian, 62–64, 63
non-Newtonian, 64
Stress tensor, 56–58, 57, 62–63, 65
for incompressible fluid, 64
Strip-theory approximation, 341
for added mass of cylinder, 151–152,
153
for added mass of spheroid, 163(8)
for aspect-ratio lifting surface, 167
heave, 375, 378–387
for lateral force on slender body,
353–354
lifting-line approach, 210
pitch, 375, 378–387
ship motions in waves, 374–397
Strouhal number, 40–41
Struik, D. J., 260
Strumming, 41, 51(13)
Subcritical case, 390, 394–395
Submarines
drag force on, 50(3)
stability of, 368
Substantial derivative, 62
423
Summation convention, 58, 143
Supercavitary propeller, 36
Supercavitating flow, lifting foil, 217,
217–218, 223–236
Supercavitating propeller, 28
Supercritical case, 390, 395–396
Supertankers
stability of, 368
wave resistance, 296
Surface tension, 14
Surface waves, definition, 247
Surge
definition sketch, 298
high-frequency limit, 310
low-frequency limit, 309, 309
Sway, 357, 357. See also Body motions
added-mass coefficient, 354
definition sketch, 298
determination, 363
high-frequency limit, 310
low-frequency limit, 309
velocity, 358, 363
Symmetric strut, 175
cavity flow past, 218–224, 219, 223,
224
Symmetry
of added-mass coefficient, 308
of damping coefficient, 308
of stress tensor, 57, 57–58
Tail fin on slender body, 343, 354
Tangential stress, 20
Tangential velocity, 162(2), 162(3)
Taylor series expansions, 149–150, 255,
289, 314, 389
Taylor wake, 35
Terminal velocity, 15, 50(2)
Theodorsen function, 237–238, 238
Thickness problem
hydrofoils, 138, 175–176, 176,
186–189
three-dimensional planar lifting
surface, 200–201
424
Thin ships, wave resistance (drag), 138
Thin-ship theory, of wave resistance,
292–294
Thin-wing theory, 292
linearized, 159
Thrust coefficient, 28, 51(7)
in self-propelled test, 34–35
Thrust-deduction coefficient, 36
Thwaites, B., 210
Time (M), 10
Time dependence, oscillatory, 236–239,
241
Todd, F. H., 19, 28
Torque, of propellers, 25–28, 27, 29
Torque coefficient, 28, 29
in self-propelled test, 34–35
Towing-tank tests, 99
Transfer function, 321–322, 334
Transport theorem, 59–61, 62, 139
Trefftz plane, 207, 208, 212
Trim, 396, 396
Trochoidal wave profile, 260–261
Troost, L., 28, 32
Tuck, E. O., 342, 382, 383, 397
Tupper, E. C., 306
Turbulence
effect on boundary layer, 91–102,
103(6)
effect on drag, 20–21, 21
effect on separation, 17
mean and fluctuating components,
88–91
of pipe flow, 70
relation to body surface, 18, 19
Uniform stream, 127
velocity potential of, 116–120,
129–131
Uniqueness proof, 116
Ünlüata, U., 263
Unsteady force, on accelerating body,
37–40, 38, 40
Ursell, F., 323
Index
Vacuum, falling bodies in, 10–11
Vacuum towing tanks, 51(9)
Van Dyke, M., 214, 215, 347
Vapor pressure, 4, 66
Velocity
terminal, 15, 50(2)
vectorial, 5
Velocity-defect law, 93, 94, 97
Velocity field, complex potential,
123–125, 124, 125
Velocity potential
complex potential, 123–125, 124, 125
definition, 117
existence in irrotational flow, 110–112
of point vortex, 177, 177
representation by Green’s theorem,
135–138
of a source, 117
for standing waves, 335(2)
stream function, 121
of uniform stream, 116–120, 129–131
Venning, E., 28
Venturi effect, 114
Virtual-mass coefficients, 156
Viscosity
of air and water, 4
coefficient of, 64
Viscosity coefficient, 5
kinematic, 65
Viscous drag, as residual drag
component, 30
Viscous drag force, 44
Viscous flow, laminar, 16, 17
Viscous fluid motion, 55–105
description of flow, 56–58
effect of stress on, 52–64
flow between two parallel walls, 67–68
transport theorem of, 59–61, 62
Viscous force, 42
on axisymmetric bodies, 47
orders of magnitude, 2–3
relation to inertial force, 42, 44
Viscous force damping, 46
Index
Viscous shear, 70
Viscous shear coefficient, 64
Viscous shear stress, 81
Viscous stress, 6–7, 37, 63–64
Viscous sublayer, 91, 98
Vortex, 40–41
complex potential, 243(12)
distribution, 177–180, 186, 187–189,
197–200
for flat plate, 180–184, 181
on unsteady hydrofoil, 233
filament, 197, 197–200, 198, 200
horseshoe, 204, 205, 205–206
in linearized lifting problem, 177–180
shed from bluff body, 40–41, 170–171
sheet, 201
bound, 201
shedding, 357
slender-body, 355, 357
trailing (free), 201, 202, 203–206,
204, 355–356
unsteady hydrofoil, 232
three-dimensional, 197–200
trailing (free), 211–212
two-dimensional, 177–180, 197
Vorticity, 110
conservation on vortex sheets, 202
trailing (free), 203–208
Vossers, G., 50
Vugts, J. H., 310, 316, 387
Wagner function, 241, 242, 323
Wake
retardation, 35
in unsteady force, 38–39
of unsteady hydrofoil, 230, 230–231,
232–236, 237
Wake fraction, 35
Walls
ideal fluid motion between, 160–161
near-wall approximation, 96, 97
parallel, flow between, 67–68, 69
Wang, S., 382
425
Ward, T. M., 296
Water, density and viscosity coefficients,
3–4
Waterplane, area and moments, 304,
371, 373, 374
Wave(s)
amplitude, 12, 13, 42–43, 50(1), 252,
260, 266, 279
relation to damping coefficient, 308
carrier, 268, 269
dimensional analysis, 12–14
dispersive, 13–14
diverging, 285, 285, 286, 290
drag, 30, 278–281, 281, 283
breaking wave–related, 286
definition, 248
in full-scale speed, 31
relation to Reynolds numbers, 33–34
of thin ships, 138
wave pattern analysis, 294–296
energy. (see energy, of waves)
equation, 389
following, 317, 377
force, on stationary bodies, 41–44
frequency, 42
front, 270, 270–271
generation by moving vessels. See Ship
waves
generation by wind, 14, 261
group velocity, 268–271, 269, 270,
272, 276–277, 283
head, 317
height, 260
average, 329–330
significant, 328–329, 331
of steepest wave, 260, 261
incident, 265, 266
linearized, 13
long-crested, 327
oblique, 265–266, 267, 318
oscillatory, 12–14
phase velocity, 12, 12
progressive, 12
426
Wave(s) (cont.)
reflected, 265, 265, 266
resistance. (see wave, drag)
short-crested, 327
side-band, 260
slender bodies in, 369–387
spectrum, 13–14, 247, 327–330, 331
relation to wind speed, 328, 329,
330, 330–331
standing, 264–266, 265
superposition, 263–268
transverse, 285, 285, 286, 290
wind-generated, 14
Wave diffraction problem, 299
Wavelength, 50(1)
linearized, 13
phase velocity of, 252, 255
relation to phase velocity, 13–14
in shallow water, 388–389
Wavemaker theory, 319, 335(6)
Wavenumber, 250, 251–252, 282, 325
in shallow water, 388–389
Wave pattern analysis, 294–296
Wehausen, J. V., 260, 261, 268, 277,
281, 293, 310, 311, 312, 316
Wetted-surface area, 29, 51(8), 51(10)
White, F. M., 88
Whitham, G. B., 277
Whitney, A. K., 222
Wiegel, R., 255, 263
Wille, R., 41
Wind, wave generation by, 14, 261
Wind speed, relation to wave spectrum,
328, 329, 330, 330–331
Wind tunnels, 50(3)
wall corrections, 161
Woods, L. C., 234, 241
Wu, T. Y., 218, 222, 239, 357
Yaglom, A. M., 95–96, 96, 97, 325, 326
Yaw, 297, 357, 357
definition sketch, 298
determination, 363
Index
high-frequency limit, 310
low-frequency limit, 309
slender body, 354
velocity, 358, 363
Yaw moment, 359, 362
cross-coupling, 360
Yeung, R. W., 342
Yih, C. S., 88, 141
Zakay, A., 32
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